Comparison Properties of Diﬀusion Semigroups on Spaces with …renesse/Docs/Dissmain.pdf ·...

Comparison Properties of Diffusion Semigroups onSpaces with Lower Curvature Bounds

Dissertation

zur Erlangung des Doktorgrades (Dr. rer. nat.)

der Mathematisch-Naturwissenschaftlichen Fakultat

der Rheinischen Friedrich-Wilhelms-Universitat Bonn

vorgelegt von Max-K. v. Renesse

aus Bochum

Bonn, Dezember 2001

Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultat der

Rheinischen Friedrich-Wilhelms-Universitat Bonn

1. Referent: Prof. Dr. Karl-Theodor Sturm

2. Referent: Prof. Dr. Werner Ballmann

Tag der Promotion:

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Intrinsic Coupling on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Central Limit Theorem for Coupled Random Walks . . . . . . 10

2.3 The Coupling Probability and Gradient Estimates . . . . . . . . . . 16

2.4 Extension to Riemannian Polyhedra . . . . . . . . . . . . . . . . . . 22

3 Alexandrov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Gradient Estimates on Alexandrov Surfaces . . . . . . . . . . . . . . . . . 36

4.1 An Integral Gauss-Bonnet-Formula . . . . . . . . . . . . . . . . . . 36

4.2 Distributional Gaussian Curvature Bounds . . . . . . . . . . . . . 46

5 The Heat Kernel on Alexandrov Spaces . . . . . . . . . . . . . . . . . . . . 49

5.1 Volume Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Dirichlet Forms and Laplacians on Metric Spaces . . . . . . . . . . 54

5.3 Laplacian Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 Heat Kernel Comparison . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5 Eigenvalue Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6 Distance and Short Time Asymptotics . . . . . . . . . . . . . . . . 65

5.7 Diffusion Process Comparison . . . . . . . . . . . . . . . . . . . . . 66

6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1 A - Remark on Coupling by Dirichlet Forms . . . . . . . . . . . . . 71

6.2 B - More about the Geometry of Alexandrov Spaces . . . . . . . . . 75

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Introduction 1

1 Introduction

The interaction between geometry and probability for stochastic processes on Riemannian

manifolds has been extensively studied since the 1950’s when Ito himself initiated the the-

ory of diffusion processes on manifolds∗. Due to the close relationship between diffusion

processes, operator semigroups and their generators it is often possible and convenient to

switch between the analytical and stochastic picture in order to develop a better under-

standing of curvature effects in diffusion semigroup theory.

In the study of stochastic processes on manifolds many authors prefer to employ the lan-

guage of stochastic differential equations including such important tools as the geometric

Ito formula and which has lead to a variety of beautiful results. However, usually the re-

quirements on the smoothness of the data are quite rigid and therefore SDE do not seem

to be the appropriate framework for stochastic analysis in particular on such non-smooth

geometries which have gained a lot of interest in recent years. On the other hand, since

the natural category of state spaces for Markov processes is formed by the measurable,

measured or topological spaces, it seems fairly plausible to use the equivalence of many

stochastic and analytical notions on smooth spaces as a basis for the definition and investi-

gation of analytical objects even on spaces with low regularity by ’doing analysis in terms

of Markov processes’. These remarks motivate the intrinsic point of view that is taken

in the present work, where certain classical results on the heat semigroup on manifolds

with lower curvature bounds are reproved or extended to either nonsmooth Riemannian

(M, g) or measured metric spaces (X, d,m) by using mainly the properties of d and m and

as little other ’extrinsic’ structure for X as possible. In particular we refrain completely

from the usage of SDE concepts whenever we deal with stochastic processes.

A first example of this approach is given in section two by a new version of Kendall’s

probabilistic proof of Yau’s gradient estimates for harmonic functions on smooth Rieman-

nian manifolds with lower (Ricci) curvature bounds. We obtain here the crucial coupling

process and coupling probability estimate from a direct by-hand-argumentation which

involves a central limit theorem for coupled geodesic random walks and an asymptotic

quadruple inequality, bypassing the sophisticated and restricted machinery of horizontal

lifts of processes to the principal frame bundle etc. Except its simplicity our method

has another major advantage to the SDE approach because it yields the coupling time

estimate irrespective if the manifold has a nonempty cut locus or not. Additionally our

proof readily suggests to be extended to more general and non-smooth spaces which is

illustrated by the example of certain Riemannian polyhedra.

∗Detailed expositions of the basic theory of stochastic analysis on manifolds can be found in themonographs [Eme89, IW89, HT94].

2 Introduction

The subsequent sections deal with comparison properties of the heat semigroup on Alexan-

drov spaces. These are geodesic metric spaces (X, d) whose lower curvature bound is

formulated in terms of the convexity of geodesic triangles and which can be considered as

a straightforward generalization of Riemannian manifolds with lower sectional curvature

bounds because many examples appear most naturally as Gromov-Hausdorff limits of se-

quences of manifolds with uniform lower sectional curvature bound. In section three we

give a short review of the most important geometric concepts and regularity properties

of Alexandrov spaces.

In section four we present a purely analytic approach to the extension of Yau’s gradient

estimate for harmonic functions onto two-dimensional Alexandrov spaces. Since Chen

and Hsu have shown that distributional lower curvature bounds are sufficient for gradient

estimates to hold we present an argument how to derive a distributional inequality for

the Gaussian curvature from Alexandrov’s geometric curvature bound, the main tools be-

ing an integral version of the Gauss-Bonnet theorem and the lower bound for the excess

measure on (X, d). A similar problem was treated in a different way by I. Nikolaev in the

case of metric spaces with two-sided curvature bounds and our discussion in this section

is inspired by his works. However, since we impose a priori a certain regularity on the

surfaces (X, d) it is not clear to which extent our results can be applied in more general

situations. It is worth mentioning in this context that our analytic proof is based upon

triangle comparison, whereas the stochastic proof of section two relies on an asymptotic

quadruple comparison property of the underlying space and that both conditions are not

equivalent in the lower curvature bound case.

The topic of section five is an investigation of the heat kernel and what we call canonical

diffusion process on Alexandrov spaces (X, d). Starting from Sturm’s construction of a

diffusion process on measured metric spaces we show that a number of classical results

about Brownian motion and the heat semigroup carry over to Alexandrov spaces, such

as the lower bound for the heat kernel by Cheeger and Yau, the short time asymptotic

formula of Varadhan and the first eigenvalue estimate for the Laplacian by Cheng. Fi-

nally we give a lower estimate for the escape speed of Brownian motion, generalizing the

well known result by Debiard, Gaveau and Mazet. Concerning the regularity of the space

(X, d) we are very general in this section, because except Alexandrov’s curvature bound

we only require an asymptotic growth condition for the volume of small geodesic balls in

(X, d) which is satisfied even in very irregular Alexandrov spaces.

As a final introductory remark we would like to recall that all comparison properties which

we have dealt with so far are usually obtained from lower Ricci curvature bounds as long

as the spaces are sufficiently smooth. On the other hand it is well justified to think of

Introduction 3

Alexandrov spaces as generalized Riemannian manifolds with lower sectional curvature

bounds and hence one might expect another class of metric spaces to be a suitable sub-

stitute for Riemannian manifolds with lower Ricci curvature bounds. Unfortunately, the

currently most promising candidates and proposals for such spaces exhibit such a little

amount of regularity or other familiar properties that a satisfactory stochastic geometric

analysis is yet to be developed before aiming at a further extension of classical comparison

results.

Acknowledgements:

I would like to express my sincerest gratitude to my scientific advisor Karl-Theodor Sturm

for giving me the opportunity to study the exciting area of stochastic differential geometry

with him, for his confidence and support. Furthermore, my thanks go to Sergio Albeverio

for creating this inspiring and cordial atmosphere in the stochastics group in Bonn and to

all its other members for the very enjoyable time, in particular to Martin Hesse and Gustav

Paulik. I am indebted also to Werner Ballmann and Anton Thalmaier for numerous

enlightening discussions, as well as to Takashi Shioya for his generous hospitality during

my stay at Tohoku University. Finally, I am grateful to my family and friends for their

long-lasting patience but most of all to Mignon for her empathy and understanding.

A remark on notation:

Throughout this work we make frequent use of Landau’s symbols, which we understand

in the following way. O(s) (or sometimes Oα,β,···(s)) is a function depending on s (and

parameters α, β, · · · ) such that O(s)/s ≤ C for s in some neighborhood of zero, o(s) has

the property that o(s)/s → 0 for s → 0 and ϑ(s) is a function for which ϑ(s) → 0 for

s→ 0 and which replaces in this text the usually more common notation O(1). In order

to distinguish from the case of a Riemannian manifold (M, g) we allocate the letters X to

a metric space and Ξ to a diffusion process. Parallel translation along a curve γ on (M, g)

is denoted by //γ and Mn,k is the simply connected n-dimensional Riemannian manifold

of constant curvature k ∈ R.

4 Intrinsic Coupling on Manifolds


The coupling method by Kendall [Ken86] and Cranston [Cra91] provides an elegant

stochastic proof of Yau’s famous gradient estimates for harmonic functions on Rieman-

nian manifolds with lower Ricci curvature bounds [Yau75]. Nevertheless from our point

of view their construction of the coupling process has two weaknesses, namely the very

’extrinsic’ concepts involved (like horizontal lifts of processes on a Riemannian manifold

onto the orthonormal frame bundle and stochastic anti-development, for instance) and

the somewhat nebulous arguments concerning the behaviour of the coupling process on

the cut locus.

In the following paragraphs we aim at a more intrinsic construction of the coupling pro-

cess on M×M which is motivated by a view towards non-smooth geometries as explained

in the introduction. Also, the construction given below easily yields the crucial coupling

probability estimate irrespective if the manifold has a nonempty cut locus or not.

2.1 Construction

The idea about coupling of Brownian Motion on a Riemannian manifold (M, g) is to

construct a stochastic process Ξ on the product M ×M such that

i) each factor Ξ1 = π1(Ξ) and Ξ2 = π2(Ξ) is a Brownian motion on (M, g)

ii) the compound process d(Ξ) of Ξ with the intrinsic distance function d on M is

dominated by a real semi-martingale ξ whose hitting time at zero TN(ξ) can be

estimated from above.

Instead of using the SDE approach we follow the lines of the Markov chain approximation

scheme for solutions to martingale problems for degenerate diffusion operators (cf. chap-

ter 11 in [SV79]). Throughout this chapter we assume that (M, g) is smooth and complete.

As a preparation we recall Jørgensen’s central limit theorem [Jør75] for geodesic random

walks: Let (M, g) be a smooth Riemannian manifold of dimension d and fix for every

x ∈M an isometry Φx : Rd '→ TxM such that the resulting function

Φ(.) : M → O(M),Φ(x) : Rd '→ TxM ∀x ∈M

is measurable. Let (ξk)k∈N be a sequence of Rd-valued and independent random variables

defined on some probability space (Ω,O, P ) whose distribution equals the normalized

Intrinsic Coupling on Manifolds 5

uniform distribution on Sd−1. A geodesic random walk (Ξε,xk )k∈N with step size ε > 0 and

starting point x ∈M is given inductively by

Ξε,x0 = x

Ξε,xk+1 = expΞε,x

k(εΦΞε,x

kξk+1),

where exp is the exponential map of (M, g). By geodesic interpolation we can extend

(Ξε,xk )k∈N to a process (Ξε,x

t )t≥0 with continuous time parameter, i.e.

Ξε,xt = expΞε,x

btc

((t− btc)εΦΞε,x

btc+1ξbtc+1

)with Ξε,x

−1 := x, ξ0 := 0 and btc := supk ∈ Z|k < t for t ∈ R. Then one may consider

two different sequences of continuous time processes obtained from rescaling Ξε,x· , namely

either by scaling the geodesic interpolation Ξxk(t) := Ξ

√1k,x

kt or by a Poisson subordination

Ξxk(t) := Ξ

√1k,x

τk(t) for k ∈ N0, where τk is a Poisson jump process on N with parameter k.

Even if Ξxk(.) is continuous by construction the cadlag process Ξx

k(.) is easier to handle

because it is a (time homogeneous) Markov process with transition function

P (Ξxk(t) ∈ A | Ξx

k(s) = y) = e−(t−s)k∑i≥0

((t− s)k)i

i!µi

1/√

k(y, A) =: (P k

t−s1‖‖A)(y)

with µε(z, A) =∫–Sd−1

z ⊂TzM1‖‖A(expz(εθ))dθ and µi = µ µ · · · µ (i times). The generator

of Ξk, or equivalently of the semigroup (P kt )t≥0, is therefore given by

Akf(x) = k(∫–

Sd−1x

f(expx(1√kθ)dθ − f(x)))

k→∞−→ ∆f(x) ∀x ∈M (2.1)

where ∆Mf(x) = 12d

trace(Hessf)(x), see lemma 5.1†. Using (2.1) and Kurtz’ semigroup

approximation theorem it is easy to show that

P kt −→PM

t for k −→∞

in the strong operator sense where PMt = et∆M

is the heat semigroup on (M, g), and

thus the weak convergence for the family Ξk· to a Brownian motion Ξx

· starting in x is

established by showing tightness of the distributions of Ξk· on the Skorokhod path space

DR+(M). Finally, the convergence of the sequence Ξk· to the same limit Ξx

· is proved by

showing that the distance between Ξkt and Ξk

t tends to zero in probability locally uniformly

with respect to t for k tending to infinity.

†Since we approximate the Laplacian on M by mean values on tangent spaces this definition of ∆ isnatural. Accordingly in this section a Brownian motion Ξ on (M, g) is a Markov process generated by∆. Of course, the normalization of ∆ is just a matter of linear time change for Ξ.


For a similar construction for two coupled Brownian motions with different starting points

x, y ∈M ×M let D(M) = (x, x) ∈M ×M |x ∈M be the diagonal in M ×M . Then

for all x, y ∈ M ×M \D(M) choose some minimal geodesic γxy : [0, 1] → M connecting

x and y. Fix a function

Φ(., .) : M ×M \D(M) → O(M)×O(M)

Φ1(x, y) := π1 Φ(x, y) : Rd '→ TxM

Φ2(x, y) := π2 Φ(x, y) : Rd '→ TyM

with the additional property that

Φ1(x, y)e1 =γxy(0)

‖γxy(0)‖,Φ2(x, y)e1 =

γyx(0)

‖γyx(0)‖if x 6= y (∗)

where ei is the i-th unit vector in Rd. On the diagonal D(M) we set

Φ(x, x) := (φ(x), φ(x)) ∈ Ox(M)×Ox(M) (∗D)

where φ : M → O(M) is some choice of bases as in the previous paragraph.

In the existence and regularity statement for a possible choice of Φ below Cut(M) ⊂M ×M is defined as the set of all pairs of points (x, y) which can be joined by at least

two distinct minimal geodesics, hence Cut(M) itself is symmetric and measurable.

Lemma 2.1. There is some choice of a minimal geodesic γxy (parameterized on [0, 1])

for each (x, y) ∈ M × M such that the resulting map γ : M × M → C1([0, 1],M),

(x, y) 7→ γxy is measurable, symmetric, i.e. γxy(t) = γyx(1 − t) for all t ∈ [0, 1], and

continuous on M × M \ (D(M) ∪ Cut(M)). Furthermore, for any measurable frame

map φ : M → Γ(O(M)) it is possible to find a measurable function Φ : M × M →O(M) × O(M) satisfying the conditions (∗) and (∗D) above and which is continuous on

M ×M \ (D(M) ∪ Cut(M)).

Proof. Suppose first that we found a measurable symmetric function γ : M × M →C1([0, 1],M) as above and let ψi ∈ Γ(O(M)), i = 1, 2 be two arbitrary continuous

frame maps on M . For (x, y) ∈ M ×M \D(M) we construct a new orthonormal frame

on TxM ⊕ TyM by Φ(x, y) = γxy/ ‖γxy‖ , ψ21, . . . , ψ

d1 , γyx/ ‖γyx‖ , ψ2

2, . . . , ψd2 out of the

frame ψ1(x), ψ2(y) via Schmidt’s orthogonalization procedure applied to the vectors

ψi(ek), k = 1, . . . , d in the orthogonal complements of γxy and γyx in TxM and TyM re-

spectively. Since the maps ∂s|s=0and ∂s|s=1

: C1([0, 1],M) → TM are continuous and the

construction of the basis Φ in TxM⊕TyM depends continuously on the data ψ1, ψ2, γxy

and γyx it is clear that the map Φ inherits the regularity properties of the function γ on

M ×M \D(M). Since D(M) is closed in M ×M and hence measurable any extension of


Φ by a measurable (φ(.), φ(.)) as above on D(M) yields a measurable map on the whole

M ×M . This proves the second part of the lemma.

Thus it remains to find a map γ as desired. In order to deal with the symmetry condition

we first introduce a continuous complete ordering ≥ on M (which can be obtained as

an induced ordering from an embedding of M into a high dimensional Euclidean space

Rl and some complete ordering on Rl) and restrict the discussion to the closed subset

D−(M) = (x, y) |x ≥ y ⊂M ×M endowed with its Borel σ-algebra which is the trace

of B(M×M) onD−(M). We define a measurable set-valued map Γ : D−(M) → 2C1([0,1],M)

as follows: for each ε > 0 choose some ε-net P ε = pεi | i ∈ N in D−(M) and choose some

minimal geodesic γpεi ,p

εjfor each pair of points pε

i , pεj ∈ P ε. Arrange the set of pairs (pε

i , pεj)

into a common sequence (pεik, pε

jk) | k ∈ N and let γε : D−(M) → C1([0, 1],M) be the

map defined inductively by

γε(x, y) = γpεi0

pεj0

for (x, y) ∈ B2ε(pεi0, pε

j0)

γε(x, y) = γpεik+1

pεjk+1

for (x, y) ∈ B2ε(pεik+1

, pεjk+1

) \k⋃

l=0

B2ε(pεil, pε

jl)

It is clear from the definition that the functions γε are measurable and, moreover, using the

geodesic equation in (M, g) together with the Arzela-Ascoli-theorem it is easy to see that

for each (x, y) ∈ D−(M) the set of curves γεxyε>0 are relatively compact in C1([0, 1],M).

Trivially any limit point of γεxyε>0 for ε tending to zero will be a minimal geodesic from

x to y. Let us choose a priori some sequence εk → 0 for k → ∞ then we define the

set valued function Γ : D−(M) → 2C1([0,1],M) for (x, y) ∈ D−(M) as the collection of all

possible limit points of Γεk(x, y), i.e.

Γ(x, y) :=

γxy

∣∣∣∣∣ ∃ subsequence εk′ and γεk′xy ∈ Γεk′ (x, y) :

γεk′xy → γxy in C1([0, 1],M) for k′ →∞

⊂ C1([0, 1],M).

The fact that we can find a measurable ’selector’, i.e. a measurable map γ : D−(M) →C1([0, 1],M) with γ(x, y) ∈ Γ(x, y) follows from a measurable selection theorem as for-

mulated in the subsequent lemma. Furthermore, the uniqueness of γxy and compactness

arguments imply that any such selector obtained from the map Γ above must be contin-

uous on D−(M) ∩ (Cut(M) ∪ (D(M))c. It is also clear that γxx is the constant curve in

x for all x ∈ M and hence we may extend our chosen γ from D−(M) continuously onto

the whole M ×M by putting γyx(t) := γxy(1− t) if (x, y) ∈ D−(M). This proves the first

assertion of the lemma and the proof is completed.

Lemma 2.2. Let (X,S) be a measurable and (Y, d) be a complete separable metric space

endowed with its Borel σ-algebra B(Y ). Let furthermore fk : X → Y be a sequence of

measurable functions which are pointwise relatively compact, i.e. for all x in X the set


fk(x)k∈N is relatively compact in Y . Let the set valued map F : X → 2Y be defined by

pointwise collecting all possible limit points of the sequence fk. Then there is a measurable

function f : X → Y with f(x) ∈ F (x) for all x ∈ X.

Proof. Since the set F (x) is obviously closed for any x in X it remains to check the

measurability of F , i.e. we need to show that F−1(O) := x|F (x)∩O 6= ∅ is measurable

in X for any O ⊂ Y open. Since any open O ⊂ Y can be exhausted by countably many

set of the type Bδ(y) with δ > 0, y ∈ Y we may replace O by Bδ(y) in the condition

above. But using the pointwise compactness of the sequence fk and a diagonal sequence

argument it is easy to show that

F−1(Bδ(y)

)=⋂δ′>δ

lim supk→∞

f−1k (Bδ′(y)) .

Choosing some sequence δ′l δ we see that in fact F−1(Bδ(y)

)is measurable. Hence we

may apply the measurable selection theorem of Kuratowksi and Ryll-Nardzewski to the

function F which yields the claim.

We now take two independent sequences (ξk)k∈N and (ηk)k∈N of Rd-valued i.i.d. random

variables with normalized uniform distribution on Sd−1 and define a coupled geodesic

random walk Ξε,(x,y)

k = (Ξε,(x,y)1,k ,Ξ

ε,(x,y)2,k ) with step size ε and starting point (x, y) in M×M

inductively by

Ξε,(x,y)

0 = (x, y)

and if Ξε,(x,y)

k ∈M ×M \D(M):

Ξε,(x,y)

k+1 =(exp

π1(Ξε,(x,y)k )

[εΦ1(Ξε,(x,y)

k )ξk+1], expπ2(Ξ

ε,(x,y)k )

[εΦ2(Ξε,(x,y)

k )ξk+1])

(2.2)

if Ξε,(x,y)

k ∈ D(M):

Ξε,(x,y)

k+1 =(exp

π1(Ξε,(x,y)k )

[εφ(π1(Ξε,(x,y)k ))ξk+1], exp

π1(Ξε,(x,y)k )

[εφ(π1(Ξε,(x,y)k ))ηk+1]

)(2.3)

where πi, i = 1, 2 are the projections of M×M on the first and second factor respectively.

As before we have at least two possibilities to extend Ξε,(x,y)

k to a process with continuous

time parameter t ∈ R+, namely

i) by geodesic interpolation Ξε,(x,y)

t ,

ii) by Poisson subordination Ξε,(x,y)

τλ(t) .


In particular choosing ε = 1/√k and λ = k in ii) for k ∈ N one obtains a sequence of

Markov processes Ξk,(x,y)

t = Ξ1/√

k,(x,y)

τk(t) on M ×M with transition function

P(Ξ

k,(x,y)

(t) ∈ A×B∣∣Ξk,(x,y)

(s) = (u, v))

= e−(t−s)k∑i≥0

((t− s)k)i

i!µi

1/√

k((u, v), A×B)

where the kernel µε : M2 × B(M2) → R is given by

µε((u, v), A×B) =

∫–

Sd−10 ⊂Rd

1‖‖A(expu(εΦ1(u,v)θ))1

‖‖B(expu(εΦ2(u,v)θ))dθ

if (u, v) ∈ D(M)c

∫–

Sd−1u ⊂TuM

1‖‖A(expu(εθ))dθ ·∫–

Sd−1v ⊂TvM

1‖‖B(expv(εθ))dθ

else.

Trivially the generator of the semigroup (P t

k,(x,y)

)t≥0 induced by Ξk,(x,y)

is

Lk = k(µ1/√

k − Id).

Lemma 2.3. Let F : M ×M → R be a smooth function. Then for k →∞

LkF (u, v) −→ LcF (u, v) ∀(u, v) ∈M ×M, locally uniformly on D(M)c

where the operator Lc = LM,φc is defined by

Lc(f ⊗ g) = ∆f ⊗ g + f ⊗∆g +1‖‖D(M)c〈∇f,∇g〉Φ (2.4)

〈∇f,∇g〉Φ(x, y) :=1

d〈Φ−1

1 (x, y)∇f(x),Φ−12 (x, y)∇g(y)〉Rd

whenever F : M×M → R is of the form F = f⊗g with smooth f, g : M → R. Moreover,

for the case F = f ⊗ 1 or F = 1⊗ g one finds Lkf ⊗ 1 → ∆f ⊗ 1 and Lk1⊗ g → 1⊗∆g

locally uniformly on M ×M for k tending to infinity.

Proof. Suppose first that (u, v) ∈ D(M)c and let U be some neighborhood with (u, v) ∈U ⊂ D(M)c. Now for any (u′, v′) ∈ U the Taylor expansion of F = f ⊗ g about (u′, v′)

and the definition of the exponential map yield

f(expu′(

1√kΦ1

(u′,v′)θ))g(expv′(

1√kΦ2

(u′,v′)θ))

= f(u′)g(v′) +1√kf(u′)〈∇g(v′),Φ2

(u′,v′)θ〉Tv′M+

1√kg(v′)〈∇f(u′),Φ1

(u′,v′)θ〉Tu′M

+1

k〈∇f(u′),Φ1

(u′,v′)θ〉Tu′M· 〈∇g(v′),Φ2

(u′,v′)θ〉Tv′M

+1

2kf(u′)Hessgv′(Φ

2(u′,v′)θ,Φ

2(u′,v′)θ)+

1

2kg(v′)Hessfu′(Φ

1(u′,v′)θ,Φ

2(u′,v′)θ) + ou′,v′(

1

k)


where o(.) is a ”little o” Landau function. In fact, ou′,v′(1k) can be replaced by oU( 1

k) due

to the smoothness of the data (M, g) and the function F . Inserting this into Lk gives

Lk(F )(u′, v′) = ∆f(u′)g(v′) + f(u′)∆g(v′)

+1

d〈Φ−1

1 (u′, v′)∇f(u′),Φ−12 (u′, v′)∇g(v′)〉Rd + ϑU(

1

k)

(2.5)

with ϑU( 1k) → 0 for k →∞ because

∫–

Sd−1

〈∇f(u′),Φ1(u′,v′)θ〉Tv′M

dθ =∫–

Sd−1

〈∇g(v′),Φ2(u′,v′)θ〉Tv′M

dθ = 0

1

2

∫–

Sd−1

Hessfu′(Φ1(u′,v′)θ,Φ

2(u′,v′)θ)dθ = ∆f(u′)

1

2

∫–

Sd−1

Hessgv′(Φ2(u′,v′)θ,Φ

2(u′,v′)θ) = ∆g(v′)∫

–Sd−1

〈∇f(u′),Φ1(u′,v′)θ〉Tu′M

· 〈∇g(v′),Φ2(u′,v′)θ〉Tv′M

=1

d〈Φ−1

1 (u′, v′)∇f(u′),Φ−12 (u′, v′)∇g(v′)〉Rd .

Now if (u, v) ∈ D(M) by definition of Lk the coupling term 〈∇f,∇g〉Φ does not appear

and thus the claim is proved.

Remark 2.1. The previous proof remains the same for general smooth functions F :

M ×M → R. The characterization of Lc as above is just more instructive.

2.2 The Central Limit Theorem for Coupled Random Walks

The operator Lc has two irregular properties, one being its degeneracy, i.e. the second

order part acts only in d of the 2d directions, and the other one being the discontinuity

of the coefficients on D(M) ∪ Cut(M). Both features together cause problems for the

definition of a semigroup etLc via the Hille-Yosida theorem. Therefore we confine ourselves

to the construction of a solution Ξ to the martingale problem for Lc in a restricted sense

by showing compactness of the (laws of the) sequence of processes(Ξ

k,(x,y)

·)

kon the space

DR+(M ×M) of cadlag paths equipped with the Skorokhod topoply.

Theorem 2.1 (Coupling Central Limit Theorem). The sequence(Ξ

k,(x,y)

·)

k≥0is tight

on DR+(M ×M) and any weak limit of a converging subsequence (Ξk′,(x,y)

· )k′ is a solution

to the martingale problem for Lc in the following restricted sense: let

(Ω∞, P∞,(x,y), (Ξ∞,(x,y)

s )s≥0) = (DR+(M ×M),w- limk′→∞

(Ξk′,(x,y)

· )∗P, (πs)s≥0)


denote the canonical process on M×M induced from a limit measure w- limk′→∞(Ξk′,(x,y)

· )∗P

on DR+(M ×M) and the natural coordinate projections πs : DR+(M ×M) → M ×M ,

then for all F ∈ C∞0 (M ×M \ (D(M) ∪ Cut(M))), F = f ⊗ 1 or F = 1⊗ g with smooth

f, g : M → R the process

F (Ξ(x,y)

t )− F (x, y)−t∫

0

LcF (Ξ(x,y)

s )ds

is a P∞,(x,y) martingale with initial value 0. In particular, under P∞,(x,y) both marginal

processes (Ξ1s := π1

s)s≥0 and (Ξ2s := π2

s)s≥0 are standard Brownian motions on (M, g)

starting in x and y respectively.

Remark 2.2. We do not claim uniqueness here for the solution to the martingale problem

in the form stated above nor a Markov property. Note also that we cautiously circum-

vented the problem of the cut locus by the choice of admissible test functions F .

We will call for short any probability measure on DR+(M ×M) with the properties above

a solution to the (restricted) coupling martingale problem.

For the proof of theorem 2.1 we show that the corresponding arguments for Euclidean

diffusions carry over to the present situation with only few changes. We follow along the

lines of chapter 8 in [Dur96], extending and simplifying the results in [Jør75].

Remember that a family (Pi)i∈I ⊂ P(X) of Borel probability measures on a topological

space (X, τ) is called tight iff

∀ ε > 0∃K ⊂ X compact: infiPi(K) ≥ 1− ε.

If (X, τ) is metrizable, complete and separable then Prohorov’s theorem states that

the family (Pi)i∈I is tight if and only if it is relatively compact with respect to the

weak-∗-topology induced on P(X) from the uniformly continuous functions on (X, τ).

- When talking about tightness of stochastic process (Ξk· )k defined on probability spaces

(Ωk,Ok, Pk) and with a common state space X one actually means tightness of the

corresponding image measures (Ξk· )∗Pk on (a properly chosen and topologized subset)

of the path space XR+ , which in the present context is DR+(M × M) equipped with

the Skorokhod topology, where M × M is endowed with the standard product metric

d((x1, y1), (x2, y2)) =√d2(x1, x2) + d2(y1, y2).

The tightness of the sequence(Ξ

k,(x,y)

· )k is shown by verifying that it satisfies the condi-

tions i) and ii) of the following theorem:


Theorem 2.2 (Tightness criterion on DR+(X)). Let (X, d) be a complete and sep-

arable metric space and let (Ωn, Pn, (Ξnt )t≥0)n∈N be a sequence of cadlag processes on X.

Then the following condition is sufficient for tightness of (Ξn· )∗(PN) on DR+(X): For all

N ∈ N, and η, ε > 0 there are x0 ∈ X, n0 ∈ N and M, δ > 0 such that

i) Pn(d(x0,Ξn0 ) > M) ≤ ε for all n ≥ n0

ii) Pn(w(Ξn, δ, N) ≥ η) ≤ ε for all n ≥ n0

with the modulus of continuity w(Ξn, δ, N)(ω) := sup0≤s, t≤N|s−t|≤δ

d(Ξns (ω),Ξn

t (ω)).

Proof. This follows just as in the caseD[0,1](X) essentially from the inequality w′(x, δ,N) ≤w(x, 2δ,N) for x ∈ DR+(X) and δ ≤ 1

2N where

w′(x, δ,N) = inf0=t0<t1···<tk=N

|ti−ti−1|≥δ

maxi≤k

sups, t∈[ti,ti+1)

d(x(s), x(t))

and the characterization of compact subsets in DR+(X) by the functionals w′(x, δ,N), cf.

theorem 15.5 in [Bil68] and theorem VI.1.5 in [JS87].

First we state an auxiliary result concerning the stopping times τk,(x,y)η on (Ω,O, P )‡ given

by

τ k,(x,y)η = inf

s ≥ 0 | d(Ξ

k,(x,y)

s , Ξk,(x,y)

0 ) ≥ η

Lemma 2.4. For all compact K ⊂M ×M and η there is a Cη ≥ 0 such that ∀δ > 0

supk≥0

sup(x,y)∈K

P (τ k,(x,y)η < δ) ≤ Cηδ.

Proof. Since(Ξ

k,(x,y)

t

)(x,y)∈M×M

t≥0is a continuous time Markov process on M × M with

generator Lk for all k ∈ N and F ∈ C0(M ×M) the process

F(Ξ

k,(x,y)

t

)−

t∫0

LkF(Ξ

k,(x,y)

s

)ds

is a P -martingale with respect to the parameter t ∈ R. Furthermore, from lemma 2.3 and

the smoothness of M it follows that for K ⊂ M ×M compact one can find a family of

smooth cut of functions(F(x,y),η(., .) : M ×M → R

)(x,y)∈K

with F(x,y),η = 1 on Bη(x, y),

vanishing on B2η((x, y))c and such that

supk≥0

sup(x,y)∈K

∥∥LkF(x,y),η

∥∥∞,M×M

≤ Cη

‡The terms stopping time, martingale etc. are used with respect to the filtrations generated by Ξk,(x,y)

τk(t) .


(this follows from (2.5) for a function like F(x,y),η((u, v)) = hη(d(x, u))hη(d(y, v)) with a

smooth real cut-off function hη ∈ C3(R)) and which implies that the process(F(x,y),η

(Ξ

k,(x,y)

t

)+ tCη

)t≥0

is a submartingale starting with the initial value 1. The optional stopping theorem applied

to τk,(x,y)η ∧ δ yields

EP

[F(x,y),η

(Ξ

k,(x,y)

τk,(x,y)η ∧δ

)+ Cη(τ

k,(x,y)η ∧ δ)

]≥ 1

from which one obtains by rearrangement

Cηδ ≥ CηEP (τ k,(x,y)η ∧ δ) ≥ EP

[1− F(x,y),η

(Ξ

k,(x,y)

τk,(x,y)η ∧δ

)]≥ P (τ k,(x,y)

η ≤ δ).

Here the last inequality follows from the right continuity of Ξk,(x,y)

and the definition of

F(x,y),η.

For N, k ∈ N and η > 0 let us introduce the following stopping times τn = τk,(x,y)n,η and

functionals σ = σk,(x,y)N,η , θ = θ

k,(x,y)N of the process

(Ξ

k,(x,y)

t

)by

τ0 := 0, τ1 := τ k,(x,y)η

τn = infs > τn−1 | d(Ξ

k,(x,y)

τn−1, Ξ

k,(x,y)

s ) ≥ η

σ = minτn − τn−1

∣∣ τn−1 ≤ N

θ = supd(Ξ

k,(x,y)

s , Ξk,(x,y)

s− )∣∣ s ≤ N

Lemma 2.5. If σ

k,(x,y)N,η > δ and θ

k,(x,y)N ≤ η then w(Ξn, δ, N) ≤ 4η.

Proof. Given 0 ≤ s < t ≤ N with t− s < δ, we have either τn−1 ≤ s < τn ≤ t < τn+1, in

which case

d(Ξk,(x,y)

s , Ξk,(x,y)

t ) ≤ d(Ξk,(x,y)

s , Ξk,(x,y)

τn−1) + d(Ξ

k,(x,y)

τn−1, Ξ

k,(x,y)

τn− )

+ d(Ξk,(x,y)

τn− , Ξk,(x,y)

τn) + d(Ξ

k,(x,y)

τn, Ξ

k,(x,y)

t ) ≤ 4η

or τn ≤ s < t ≤ τn+1 which yields

d(Ξk,(x,y)

s , Ξk,(x,y)

t ) ≤ d(Ξk,(x,y)

s , Ξk,(x,y)

τn) + d(Ξ

k,(x,y)

τn, Ξ

k,(x,y)

t ) ≤ 2η

Proof of theorem 2.1. Tightness. For given compact K ⊂ M ×M , (x, y, ) ∈ K, we may

assume w.l.o.g. that the sample paths of all(Ξ

k,(x,y)

·)

k∈N lie entirely in B1/√

k(K), because

otherwise we stop each process when it leaves K and let K M ×M in a final step.


Since the condition i) of theorem 2.2 is trivially satisfied, by lemma 2.5 it is sufficient

to show that for all N ∈ N, and η, ε > 0 one can find δ > 0 and k0 ∈ N such that

P (σk,(x,y)N,η > δ; θ

k,(x,y)N ≤ η/4) ≥ 1− ε for all k ≥ k0.

First note that trivially P (θk,(x,y)N > η) = 0 for k ≥ 1/η2 because

d(Ξk,(x,y)

s , Ξk,(x,y)

s− ) ≤√

1

k(τk(s)− τk(s−)) ≤

√1

kP -a.s. (2.6)

and we may assume w.l.o.g. that τk is a regular Poisson process with jumps of size 1. Thus

it remains to verify that it is possible to find δ > 0 and k0 such that P (σk,(x,y)N,η ≤ δ) ≤ ε

for all k ≥ k0. For this introduce the functional Lk,(x,y)N,η = maxl|τ k,(x,y)

l,η ≤ N. Lemma

2.4 and Fubini’s theorem yield

EP (exp (−τ k,(x,y)η )) =1−

∫R+

(1− P (τ ≤ s))e−sds ≤ 1−

12Cη∫0

(1− Cηs)e−sds

≤ 1− 1

2

12Cη∫0

e−sds =: λη < 1 ∀ (x, y) ∈ K, k ∈ N

and by the strong Markov property supk sup(x,y)∈K EP (exp (−τ k,(x,y)l,η )) ≤ λl

η. Consequently

for all k ∈ N, (x, y) ∈ K

P (Lk,(x,y)N,η ≥ l) = P (τ

k,(x,y)l,η ≤ N) ≤ eNEP (exp (−τ k,(x,y)

l,η )) ≤ eNλlη

and thus finally for any δ > 0, l ∈ N

P (σk,(x,y)N,η ≤ δ) ≤ P (σ

k,(x,y)N,η ≤ δ;L

k,(x,y)N,η ≤ l) + P (L

k,(x,y)N,η ≥ l)

≤ l sup(u,v)∈K

P (τ k,(u,v)η ≤ δ) + eNλl

η

≤ lCηδ + eNλlη

by lemma 2.4. Hence for N, η, ε given choose l0 ∈ N such that eNλl0η ≤ ε/2 and δ > 0 such

that l0Cηδ ≤ ε/2. Then choosing k0 ≥ 2/η2 establishes condition ii) and the tightness

assertion is proved.

Martingale Problem for Lc. Since Lk generates the process(Ξ

k,(x,y)

·)

this is also true for

its realization on the path space (DR+(M ×M), (Ξk,(x,y)

· )∗P, (πs)s≥0) and which is in this

case equivalent to

F (πt)− F (x, y)−t∫

0

LkF (πs)ds is a nomalized (Ξk,(x,y)

· )∗P -Martingale (2.7)


for all F ∈ Dom(Lk) ⊃ C30(M ×M). If F ∈ C∞

0 (M ×M \ (D(M) ∪Cut(M))) by lemma

2.3∥∥LkF − LcF

∥∥∞ → 0 for k → ∞ and thus by a general continuity argument (cf.

lemma 5.5.1. in [EK86]) we may pass to the limit in the statement above provided k′ is

a subsequence such that w- limk′→∞(Ξk′,(x,y)

· )∗P exists. Finally, the assertion concerning

the marginal processes follows either from the results [Jør75] or from putting F = 1⊗ f

and F = f ⊗1 in (2.7) respectively, in which case one may pass to the limit for k′ tending

to infinity without further restriction on the support of f .

For the derivation of the coupling time estimate it is easier to work with the (continuous)

interpolated processes (Ξk,(x,y)

· )k as approximation of a suitable limit Ξ(x,y)

· . Therefore we

need the following

Corollary 2.1. The sequence of processes Ξk,(x,y)

· is tight. For any subsequence k′ the

sequence of measures (Ξk′,(x,y)

· )∗P on DR+(M ×M) is weakly convergent if and only if

(Ξk′,(x,y)

· )∗P is, in which case the limits coincide. In particular the family ((Ξk,(x,y)

· )∗P )k

is weakly precompact and any weak accumulation point is a solution of the (restricted)

coupling martingale problem, which is supported by CR+(M ×M).

Proof. By construction of (Ξk,(x,y)

· ) and (Ξk,(x,y)

· ) we have Ξk,(x,y)

t = Ξk,(x,y)1kτk(s) for all t ≥

0, k ∈ N, i.e.

(Ξk,(x,y)

· ) = (Ξk,(x,y)

· ) Θk

with the random time transformation Θk(s, ω) = 1kτk(s, ω). Moreover, every process

(Ξk,(x,y)

· ) has continuous paths, so that

supp

(w- limk′→∞

(Ξk′,(x,y)

· )∗P

)⊂ CR+(M ×M)

for every possible weak limit of a converging subsequence (Ξk′,(x,y)

· )∗P and since the se-

quence Θk converges to IdR+ weakly, the continuity argument in section 17. of [Bil68] can

be applied, giving

w- limk′→∞

(Ξk′,(x,y)

· )∗P = w- limk′→∞

(Ξk′,(x,y)

· )∗P (2.8)

for that specific subsequence, i.e. we have shown thatfor any subsequence k′ →∞((Ξ

k′,(x,y)

· ) ⇒ ν

)=⇒

((Ξ

k′,(x,y)

· ) ⇒ ν

).

To prove the other implication note first that (2.6) implies the almost sure continuity of

the coordinate process π· w.r.t. any weak limit of (Ξk,(x,y)

· )∗P . We may also write

(Ξk,(x,y)

· ) Θk = (Ξk,(x,y)

· ) Θk(Θk)


where Θk(s, ω) = inft ≥ 0 | t > Θk(s, ω) is the (right continuous) generalized upper

inverse of Θk which converges to IdR+ weakly, too. Since∥∥Θk(Θk)− IdR+

∥∥∞ ≤ 1

kand

(Ξk,(x,y)

· ) is a continuous process, it is easy to see that (Ξk,(x,y)

· ) is tight on CR+(M ×M) if

and only if (Ξk,(x,y)

· ) Θk(Θk) is tight on DR+(M ×M) and thus we may argue as before.

Finally, the compactness itself comes from theorem 2.1 as well as as the characterization

of any limit as a solution to the (restricted) coupling martingale problem via equation

(2.8).

2.3 The Coupling Probability and Gradient Estimates

The curvature condition on the Riemannian manifold (M, g) enters our probabilistic proof

of gradient estimates through the following lemma, in which we confine ourselves to the

only nontrivial case of strictly negative lower (sectional) curvature bounds.

Lemma 2.6. Let (Md, g) be a smooth Riemannian manifold with SecM ≥ −k, k > 0

and let x, y ∈ M , x 6= y, be joined by a unit speed geodesic γ = γxy. Then for any

ξ, η ∈ Sd−1x ⊂ TxM

d(expx(tξ), expy(t//γη)) ≤ |xy|+ t〈η − ξ, γ(0)〉TxM

+1

2t2

√k

sk(|xy|])((|ξ⊥|2 + |η⊥|2)ck(|xy|)− 2〈η⊥, ξ⊥〉TxM

)+ oγ(t

2) (2.9)

where //γ denotes parallel translation on (M, g) along γ and ξ⊥, η⊥ denote the normal

(w.r.t. γ(0)) part of ξ and η respectively. In particular if ξ⊥ = η⊥ and ξ‖

= −η‖

d(expx(tξ), expy(t//γη)) ≤ |xy| − 2t〈ξ, γ(0)〉TxM + t2√k|ξ⊥|2 + oγ(t

2). (2.10)

Proof. If y is not conjugate to x along γ then the assertion of the lemma is derived by

the standard argument: for sufficiently small t there is a unique geodesic γt close to γ

connecting expx(tξ) with expy(t//γη) and which depends smoothly on t in a neighborhood

of zero. In particular γt → γ for t→ 0 and the first and second variation formula for the

arclength of γt yield

L(γt) = |xy|+ t〈η − ξ, γ(0)〉TxM +1

2Iγ(J

⊥ξ,η, J

⊥ξ,η)t

2 + oγ(t2).

Here J⊥ξ,η is the (w.r.t. γ) normal part of the Jacobi-field induced from the geodesic

variation c(t, s) = γt(s) and Iγ is the index form of (M, g) along γ. Choose some parallel

o.n. basis (ei)i=1,...,d, e1 = γ along γ and define the vector field V ⊥ξ,η along γ by

V ⊥ξ,η(s) =

d∑i=2

(ξick(t) +

ηi − ξick(|xy|)sk(|xy|)

sk(t)

)ei(s) (2.11)


then we have that J⊥ξ,η(0) = J⊥ξ,η(0) and J⊥ξ,η(|xy|) = J⊥ξ,η(|xy|) such that the index lemma

gives Iγ(J⊥ξ,η, J

⊥ξ,η) ≤ Iγ(V

⊥ξ,η, V

⊥ξ,η). By the definition of the index form and the curvature

assumption one finds that Iγ(V⊥ξ,η, V

⊥ξ,η) ≤ Iγ(V

⊥ξ,η, V

⊥ξ,η), where γ is a unit speed geodesic

of arclength |xy| in in the model space Md,k and V⊥ξ,η is a vector field over γ defined by

(2.11) with ei replaced by an analogous parallel o.n. basis ei along γ. But since in this

case V⊥ξ,η is a Jacobi field in Md,k over γ one finds

Iγ(V⊥ξ,η, V

⊥ξ,η) = 〈V ⊥

ξ,η,∇γV⊥ξ,η〉∣∣∣|xy|

0

=

√k

sk(|xy|)

d∑i=2

[(η2

i + ξ21)ck(|xy|) + ηiξi

(s2

k(|xy|)− c2k(|xy|)− 1)]

=

√k

sk(|xy|)((|ξ⊥|2 + |η⊥|2)ck(|xy|)− 2〈η⊥, ξ⊥〉TxM

)In the case that y is conjugate to x along γ we may find a partition 0 = s0 < s1 < · · · <sn = |xy| such that xi+1 = γ(si+1) is not conjugate to xi = γ(si) along γi = γ|[si,si+1].

Let V ⊥ξ,η be the vector field along γ be defined by (2.11) as before and for each i define

xit = expxi

(tV ⊥ξ,η(si)) and for sufficiently small t let γi

t be the unique geodesic close to

γi connecting xit with xi+1

t which give rise to a Jacobi-field J iξ,η on each γi. Since by

construction we have that J i,⊥ξ,η and V ⊥

ξ,η coincide at the endpoints of each interval [si, si+1]

we may apply the same arguments as before to each of the segments γi from which we

deduce

d(expx(tξ), expy(t//γη)) ≤n−1∑i=0

d(xit, x

i+1t )

≤n−1∑i=0

d(xi, xi+1) + t〈η − ξ, γ(0)〉TxM +1

2t2

n−1∑i=0

Iγi(J i,⊥

ξ,η , Ji,⊥ξ,η ) + on,γ(t

2)

≤ |xy|+ t〈η − ξ, γ(0)〉TxM +1

2t2

n−1∑i=0

Iγi(V

i,⊥ξ,η , V

i,⊥ξ,η ) + on,γ(t

2)

with γ and V⊥ξ,η as before, the upper (or lower) index i meaning its restriction to the

interval [si, si+1] and thus

= |xy|+ t〈η − ξ, γ(0)〉TxM +1

2t2Iγ(V

⊥ξ,η, V

⊥ξ,η) + on,γ(t

2)

from which one obtains the assertion of the lemma as in the previous case. The second

statement of the lemma follows from (2.9) and the fact that cosh(t)− 1 ≤ sinh(t) for all

nonnegative t.

Remark 2.3. The proof of lemma 2.6 shows that the error term oγ(t2) in (2.9) may

be replaced by a uniform error estimate o(t2) as long as x 6= y range over a compact

K ⊂M ×M \D(M).


Proof. By the smoothness assumption on (M, g) the injectivity radius of (M, g) is locally

uniformly bounded away from zero and hence for x, y ∈ M sufficiently close and t suf-

ficiently small the geodesic variation γt of γ0 = γxy which connects xt = expx(tξ) with

yt = expy(tη) is uniquely given by γt(s) = expxt(s logxt

yt), which is smooth in t, x, y ∈Mand ξ ∈ TxM, η ∈ TyM for small t, for x, y sufficiently close and ξ, η bounded. Since the

arclength functional L : C∞([0, 1],M) → R+ is also smooth we find that for any smooth

local frame map φ ∈ Γ(M,O(M)) the function

L : [0, ε)×M ×M × Rd × Rd → R+ L(t, x, y, u, v) = L(γt)

with γt as above, where ξ = φx(u) and η = φy(v), is smooth in some neighbourhood of

0×x = y×(0, 0). From this it follows that in fact the error term in (2.9) is uniform

(for ξ, η ∈ TxM, ‖ξ‖ = ‖η‖ = 1) as long as x and y range over a compact set sufficiently

close to the diagonal in M ×M . If x and y range over a general compact set K ⊂M ×Mwe repeat the subpartitioning argument of the previous lemma along some geodesic γxy.

It follows then from the locally uniform lower bound for the injectivity radius that the

total number of necessary subdivisions of γxy is uniformly bounded for x, y ∈ K which

establishes the claim.

In order to apply the previous lemma to the the sequence (Ξl,(x,y)

. )l we require in addition

to condition (∗) on page 6 that Φ(., .) satisfies

Φ2(u, v) Φ−11 (u, v) = //γuv

on (TuM)⊥γxy ∀ (u, v) ∈M ×M \D(M) (∗∗)

where γuv corresponds to some (w.r.t. u, v ∈ M × M \ D(M) symmetric) choice of

connecting unit speed geodesics. A function Φ(., .) satisfying (∗) and (∗∗) realizes the

coupling by reflection method (cf. [Ken86, Cra91]) in our present context. In order to

see that we actually may find at least one such map Φ(., .) which is also measurable

we may proceed similarly as in the proof of lemma 2.1: from a given measurable and

symmetric choice γ.. : M ×M → C1([0, 1],M) and a continuous frame ψ ∈ Γ(O(M))

we obtain Φ1(x, y) by an appropriate rotation of ψ(x) such that Φ1(x, y)e1 = γxy/ ‖γxy‖.Φ2(x, y) is then obtained from Φ1(x, y) by parallel transport and reflection w.r.t. the

direction of γxy. Since these operations depend continuously (w.r.t. to the C1-norm)

on the curve γxy, Φ(., .) inherits the measurability and continuity properties of the map

γ.. : M ×M → C1([0, 1],M). For δ > 0 let us introduce the functional TD,δ on the path

space of M ×M by

TD,δ : CR+(M ×M) → R ∪ ∞, TD,δ(ω) = infs ≥ 0 | d(ω1s , ω

2s) ≤ δ.

with ωis = πi(ωs), i = 1, 2 being the projections of the path ω onto the factors. Then the

coupling time TD = TD,0 is the first hitting time of the diagonal D(M) ⊂M ×M .


Theorem 2.3 (Coupling Probabilty Estimate). Let SecM ≥ −k, , k > 0 and let

Φ be chosen as above. Then for arbitrary x, y ∈ M and any weak limit P∞,(x,y) =

w- liml′→∞

(Ξl′,(x,y)

· )∗P on CR+(M ×M) the following estimate holds true:

P∞,(x,y)(TD = ∞) ≤ d− 1

2

√k d(x, y).

Proof. For x = y there is obviously nothing to prove. So let (x, y) 6∈ D(M) and assume

first that M is compact. For δ > 0 let

T lD,δ : Ω → R ∪ ∞, T l

D,δ(ω) = inft ≥ 0 | d(Ξl,(x,y)

s (ω)) ≤ δ = TD,δ (Ξl,(x,y)

· )(ω)

be the first hitting time of the set Dδ = (x, y) ∈ M ×M | d(x, y) ≤ δ for the process

(Ξl,(x,y)

s )s≥0, where (Ω,O, P ) is the initial probability space on which the random i.i.d.

sequences (ξ)i∈N (and (ηi)i∈N) are defined. By the choice of Φ for (u, v) ∈ D(M)c, θ ∈Sd−1 ⊂ Rd and

(uε, vε) = (expu(εΦ1θ), expv(εΦ2θ))

we obtain from lemma 2.6 and remark 2.3

d(uε, vε) ≤ d(u, v)− 2ελ+√kε2χ+ o(ε2)

where λ = pr1θ is the projection of θ onto the first coordinate axis and χ =∥∥θ⊥∥∥2

Rd is

the squared length of the orthogonal part of θ. This estimate inserted into the inductive

definition of (Ξl,(x,y)

t )t≥0 yields in the case Ξ1√l,(x,y)

bltc ∈ D(M)c

d(Ξl,(x,y)

t ) = d(Ξ1√l,(x,y)

lt ) ≤ d(Ξ1√l,(x,y)

bltc )− 2

(lt− bltc√

l

)λbltc+1

+√k

(lt− bltc√

l

)2

χbltc+1 + o

[(lt− bltc√

l

)2]

with the random variables λi = pr1ξi and χi =∥∥ξ⊥i ∥∥2

Rd , from which one deduces by

iteration

≤ d(x, y)− 21√l

bltc∑i=0

λi+1 − 2

(lt− bltc√

l

)λbltc+1 +

√k1

l

bltc∑i=0

χi+1

+√k

(lt− bltc√

l

)2

χbltc+1 + bltco(lt− bltc√

l

)2

= d(x, y)− 2Slt +

√k1

l

bltc∑i=0

χi+1 + ρt(l) =: rl(t) (2.12)

at least on the set T lD,δ > t, with

Slt :=

1√lSlt, St := Sbtc + (t− btc)Sbtc+1, Sk :=

k∑i=0

λi


and

ρt(l) → 0 for l→∞.

Define furthermore the stopping times

T lδ : Ω → R ∪ ∞, T l

N = inft ≥ 0 | rlt ≤ δ

then the inequality above implies T lD,δ > n ⊂ T l

δ > n for all l, n ∈ N and hence

PP,Ω(T lδ > n) ≥ PP,Ω(T l

D,δ > n) =

∫TD,δ>n

(Ξl,(x,y)

· )∗(P )(dω) (2.13)

where the second integral is taken on a subset of the path space Ω′ = C(R+,M×M) with

respect to the image measure of P under (Ξl,(x,y)

· ). By assumption we have P∞,(x,y) =

w- liml′→∞

(Ξl′,(x,y)

· )∗P and the lower semi-continuity of the function TD,δ w.r.t. to the topol-

ogy of locally uniform convergence on the path space implies that the set TD,δ > n ⊂C(R+,M ×M) is open. Thus from (2.13) it follows that

PP∞,(x,y)(TD,δ > n) =

∫TD,δ>n

(Ξ∞,(x,y)

· )∗(P )(dω)

≤ lim infl′→∞

∫TD,δ>n

(Ξl′,(x,y)

· )∗(P )(dω) ≤ lim infl′→∞

PP,Ω(T l′

δ > n) = P(Tδ(r∞) > n).

The last equality is a consequence of Donsker’s invariance principle applied to the sequence

of processes (rl·)l∈N: since each (λi)i and (χi)i are independent sequences of i.i.d. random

variables on Ω, P,A with

E(λi) = 0, E(λ2i ) =

√1

d, E(χi) =

d− 1

d

one finds that (rl·)l converges weakly to the process r∞· with

r∞t = d(x, y) +2√dbt +

√kd− 1

dt (2.14)

such that in particular PP,Ω(Tδ(r∞) = n) = 0 and we can pass to the limit in the last

term on the right hand side of (2.13). Letting n tend to infinity leads to

PP∞,(x,y),Ω′(TD,δ = ∞) ≤ PP,Ω(Tδ(r∞) = ∞),

where δ > 0 was chosen arbitrarily from which we finally may conclude

PP∞,(x,y),Ω′(TD = ∞) ≤ PP,Ω(T0(r∞) = ∞). (2.15)


with T0 being the first hitting time of the origin for the semi-martingale r∞. Using a

Girsanov transformation of (Ω,O, P ) ([KS88], sec. 3.5.C) the probability on the right

hand side can be computed precisely to be

PP,Ω(T0(r∞) = ∞) = 1− e−

12

√k(d−1)d(x,y) ≤ d− 1

2

√k d(x, y),

which is the claim in the compact case. For noncompact M we choose some open pre-

compact A ⊂ M such that (x, y) in K. We may stop the processes (Ξl′,(x,y)

· ) when they

leave A and repeat the previous arguments for the the stopping time TA,D,δ = TD,δ ∧ TAc

with TAc = infs ≥ 0 |ωs ∈ Ac which gives instead of (2.13)

PP,Ω(T lδ ∧ TAc > n) ≥

∫TD,δ∧TAc>n

(Ξl,(x,y)

· )∗(P )(dω)

From this we obtain (2.15) if we successively let tend l → ∞, A → M ×M , δ → 0 and

n→∞.

Remark 2.4. In the proof of theorem 2.3 we did not use that P∞,(x,y) is (necessarily)

a solution of the restricted coupling martingale problem. In fact, theorem 2.1 is not

involved at all at this stage, except that it guarantees the existence of a weakly converging

subsequence (Ξl′,(x,y)

· )∗P .

Remark 2.5. Also in the case of lower Ricci curvature bounds the same type of argu-

ments should yield the extension of theorem 2.3. However, the difficulties arise from the

fact that lower Ricci bounds lead to a uniform upper estimate of the expectation of the χi

in (2.12) only. Since these random variables are also only asymptotically mutually inde-

pendent, one has to find and apply an appropriate central limit theorem to the expression1l

∑bltci=0 χi+1 in order to obtain the pathwise(!) upper bound for the distance process by

the semimartingale (2.14).

For different (local and global) versions of the following result as well as for extensions to

harmonic maps see the papers by W. Kendall. We state the global version only because

it is easiest to formulate whereas its stochastic proof received most scepticism due to

non-differentiability of the distance function on the cut locus.

Corollary 2.2 (Gradient estimate for harmonic functions). If u is a harmonic,

nonnegative and uniformly bounded function on M , then

|u(x)− u(y)| ≤ ‖u‖∞ (d− 1)√k d(x, y) (2.16)

Proof. From elliptic regularity theory we now that u ∈ C∞(M). Let x 6= y be given.

Since ∆u = 0 we find Lc(u ⊗ 1) = Lc(1 ⊗ u) = 0 and from theorem 2.1 it follows that

both processes ((u ⊗ 1)(πs))s and ((1 ⊗ u)(πs))s are nonnegative continuous bounded


martingales with respect to the probability measure P∞,(x,y), where π· = (π1, π2)· is the

projection process on the path space CR+(M ×M). For any s > 0 we obtain by means of

the optional stopping theorem

u(x)− u(y) = (u⊗ 1)(π0)− (1⊗ u)(π0)

= EP∞,(x,y) [(u⊗ 1)(πs∧TD)− (1⊗ u)(πs∧TD

)]

which equals, since (u⊗ 1)(πTD) = (1⊗ u)(πTD

) on TD <∞

= EP∞,(x,y) [((u⊗ 1)(πs∧TD

)− (1⊗ u)(πs∧TD))1‖‖s<TD]

≤ 2 ‖u‖∞ PP∞,(x,y)(TD ≥ s)

Passing to the limit for s→∞ and using theorem 2.3 proves the claim.

2.4 Extension to Riemannian Polyhedra

As indicated in the introduction the stochastic proof of gradient estimates given above

suggests to be extended to certain non-smooth spaces since it depends essentially only

on two facts, namely the central limit theorem for the factor processes (πi(Ξk. )) and the

asymptotic quadruple estimate (2.10). Therefore one might think of situations where the

same kind of argument applies.

Definition 2.1. Let X be an n-dimensional topolgical manifold equipped with a complete

metric d. We call (X, d) an n-dimensional Riemannian polyhedron with lower curvature

bound κ ∈ R if X =⋃

i Pi is obtained by gluing together (in a locally finite fashion)

convex closed patches Pi ⊂ Mni (i ∈ I) of n-dimensional Riemannian manifolds with

uniform lower sectional curvature bound κ along their boundaries, where

i) the boundary ∂Pi =⋃

j Sij ⊂ Mi of each Pi ⊂ Mi is the union of totally geodesic

hypersurfaces Sij in Mi

ii) each Sij ⊂ X is contained in the intersection of at most two Pk ⊂ X and Sij ⊂ Mi

where Skl ⊂ Mk are isometric whenever two adjacent patches Pi ⊂ X and Pk ⊂ X

have a common face Sij ' Skl ⊂ (X, d)

iii) the sum of the dihedral angles for each face of codimension 2 is less or equal 2π.

Examples 2.1. The boundary ∂K of a convex Euclidean polyhedron K ⊂ Rn (with

nonempty interior K) is a (n-1)-dimensional Riemannian polyhedron with lower curvature

bound 0 in our sense. A simple example for the case κ < 0 in two dimensions is the surface

of revolution obtained from a concave function f : [a, b] → R+, f ∈ C1[a, b] ∩ C2([a, c) ∪(c, b]) with c ∈ (a, b) and

f ′(c) = 0 and f ′′/f =

−k2

1 on [a, c)

−k22 on (c, b].


Constructions: The 2π-condition iii) above assures that (X, d) is an Alexandrov space

with Curv(X) ≥ κ (see section 3), and we can use the result in [Pet98] that a geodesic

segment connecting two arbitrary metrically regular points does not hit a metrically sin-

gular point, i.e. a point whose tangent cone is not the full Euclidean space. Moreover,

from condition ii) it follows in particular that metrically singular points can occur only

inside the (n− 2)-skeleton Xn−2 of X. Thus there is a natural parallel translation along

any geodesic segment γxy whenever x and y are regular and which is obtained piecewisely

from the parallel translation on the Riemannian patches Pi and from the natural gluing

of the tangent half-spaces for points x ∈ Xn−1 \ Xn−2 lying on the intersection of two

adjacent (n-1)-faces Sij ' Skl ⊂ X. Similarly we can define the exponential map expx for

every regular point x ∈ X, i.e. for given ξ ∈ TxX we obtain a unique (”quasi-geodesic”)

curve R+ 3 t→ expx(tξ) (and which can be represented as a union of geodesic segments

on the patches Pi). With these constructions at our disposal we can verify a non-smooth

version of the asymptotic quadrangle estimate of lemma 2.6:

Proposition 2.1. Let (X, d) be a n-dimensional Riemannian polyhedron with lower cur-

vature bound κ ∈ R and let x, y ∈ X \ Xn−2 be connected by some segment γxy. Then

for ξ ∈ TxX, ‖ξ‖ = 1 the estimate (2.10) holds, where the error term o(t2) can be chosen

uniform if x 6= y range over a compact subset of X \Xn−2.

Proof. Let us prove (2.10) for fixed x, y ∈ X \ Xn−2 and ξ ∈ TxX first, i.e. without

addressing the problem of uniformity. Suppose furthermore that for some Pi we have

γxy ⊂ Pi, i.e. γxy is entirely contained in the (closed) patch Pi, then we distinguish three

cases:

i) If γxy ⊂ Pi then due to lemma 2.6 there is nothing left to prove.

ii) x ∈ Pi and y ∈ Pi ∩ Pj for some j. Since y is assumed to be regular the first order

part of estimate (2.10) is obviously true and we may focus on the second order part

which corresponds to orthogonal variations of the geodesic γ, i.e. we may assume

that ξ = ξ⊥ in (2.10). If γxy is orthogonal to the hypersurface ∂Pi ∩ ∂Pj at y or x

and y are both in Pi ∩ Pj then again there is nothing to prove since we in this case

we have to consider geodesic variations which take place completely on one of the

patches Pi ⊂Mi or Pj ⊂Mj and we can apply lemma 2.6 on Mi or Mj respectively.

Consequently we only have to treat the case that γxy is neither parallel nor orthog-

onal to ∂Pi ∩ ∂Pj, i.e. 0 < 〈 γxy(|xy|)‖γxy(|xy|)‖ , ν〉TyMi

< 1 where ν denotes the outward unit

normal vector of ∂Pi.

Let η = //γxyξ be the parallel translate of a unit


vector ξ ∈ TxMi normal to γxy. Then ζ = η − 〈ν,η〉〈ν,γxy(|xy|)〉 γxy(|xy|) ∈ TyMi ∩ TyMj

is the unique vector in the intersection of the γxy(|xy|), η-plane and the tangent

hyperplane to ∂Pi in y which is determined by its w.r.t. γxy orthogonal projection

η. For its length we obtain ‖ζ‖ = sin−1 α where α is the angle enclosed by γxy and

η at y. Since ∂Pi ⊂ Mi and ∂Pj ⊂ Mj are totally geodesic the point z = expy(tζ)

also lies on ∂Pi ∩ ∂Pj and the triangle inequality yields

dX(expx(tξ), expy(t//γxyξ)) ≤ dMi

(expx(tξ), z) + dMj(z, expy(tη)) (2.17)

where dX , dMiand dMj

denote the distance functions on X, Mi and Mj respectively.

Now the estimate (2.9) of lemma 2.6 applied to ξ and ζ in Mi yields

dMi(expx(tξ), expy(tζ)) ≤ dMi

(x, y)− tcosα

sinα+ t2

√k|ξ⊥|2 + oγ(t

2)

since trivially 〈γ, ζ〉 = cos αsin α

and by construction ζ⊥ = η = //γxyξ. As for the distance

dMj(z, expy(tη)) remember that by the smoothness assumption of the curvature of

Mj is locally uniformly bounded and from the Toponogov triangle comparison and

the cosine formula on the model spaces Md,κ we may infer with β = ^TyMj(ζ, η)

dMj(z, expy(tη) = t

√|ζ|2 + 1− |ζ| cos β + o(t2) = t

cosα

sinα+ o(t2)

because all vectors γxy, η and ζ lie on a common hyperplane and as η ⊥ γ we have

α = π/2− β. Inserting the the last two inequalities into (2.17) yields (2.10).

iii) x ∈ Pi ∩ Pk and y ∈ Pi ∩ Pj. We may argue similarly as in ii) by subdividing the

quadruple into two geodesic triangles on Mj and Mk and a remaining quadruple on

Mi. - Alternatively, if z ∈ γxy ∩ Pi 6= ∅ then one may subdivide γxy = γxz ∗ γzy and

argue as in ii). If γxy ∩ Pi = ∅, then again we have to deal with variations of γxy on

a single Riemannian patch Pk only, where Pk depends on the direction ξ⊥γxy.

The discussion above proves (2.10) when γxy ⊂ Pi for some Pi. In the general case

when γxy is not contained in a single patch we subdivide γxy = γi1 ∗ γi2 ∗ · · · ∗ γim into

pieces γik ⊂ Pik lying entirely on one of the patches which we consider separately: let

x1, . . . , xm = γxy ∩Xn−1 be the set of (transversal) intersections of γxy and Xn−1 and

for each k = 1, . . . ,m, t > 0 let zkt = expxk

(tzk) where the direction zk (depending on

the initial direction ξ ∈ TxX) is chosen as in ii). As before the triangle inequality yields

the simple upper bound d(xt, yt) ≤ d(xt, z1t ) + d(z1

t , z2t ) + · · · + d(zm−1

t , zmt ) + d(zm

t , yt)

for the distance between xt = expx(ξt) and yt = expy(t//γxyη). On each patch Pik we

may apply the previous discussion i) - iii) in order to derive asymptotic estimates for

d(xt, z1t ), d(z

1t , z

2t ), . . . , d(z

m−1t , zm

t ) and d(zmt , yt), where it is important to note that for

sufficiently small t the variations ηkt of the pieces of γik which we construct on each

patch Pik also lie entirely on Pik . (This follows from the fact that the segment γxy lies


at a strictly positive distance away from Xn−2 which comprises the set of points where

more than just two patches intersect.) Hence we obtain an upper bound for the distance

d(xt, yt) in the global quadruple by a sum of distances d(xt, z1t ), d(z

1t , z

2t ), d(z

m−1t , zm

t )

and d(zmt , yt) in geodesic quadrangles and triangles which are each entirely contained

in a single patch. Analogously to the final step in ii) summing up the corresponding

asymptotic upper estimates we recover (2.10) for the global quadruple due to the special

choice of the directions zk|k ∈ 1, . . . ,m.Finally, the uniformity assertion is obtained in a similar way by combining the arguments

in lemma 2.3 on each patch Pi with the observation that for a given compact set K ⊂X \Xn−2 the collection of all segments γxy| (x, y) ∈ K×K also lies at a strictly positive

distance away fromXn−2, which may be inferred from a simple compactness consideration.

This implies that there is some t0 > 0 such that for all t ≤ t0 and x, y ∈ K all variations

γxy,t constructed in the previous paragraph determine a well-defined sequence of geodesic

triangles and quadrangles located on the individual patches as above. Hence, by the

smoothness of the patches (and the fact that only finitely many patches are involved for

x, y ∈ K ×K) we may conclude precisely as in lemma 2.3 that the estimate (2.10) is in

fact locally uniform in the sense stated above.

As a second preparation for the probabilistic approach to a gradient estimate on (X, d) we

need to state precisely what we understand by a Brownian motion in the present situation.

Definition 2.2. The (’Dirichlet-’)Laplacian ∆X on (X, d) is defined as the generator of

the Dirichlet form (Ec, D(E)c) which is obtained as the L2(X, dm =∑

i dmibPi)-closure

of the classical energy form E(f, f) =∑

i

∫Pi|∇f |2 dmi on the set of Lipschitz functions

on (X, d) with compact support. A continuous Hunt process whose transition semigroup

coincides with the semigroup associated to (Ec, D(Ec)) on L2(X, dm) is called a Brownian

motion on (X, d).

Lemma 2.7. Let A denote the set⋂

iC∞(Pi)∩Lip(X)∩Cc(X\Xn−2) of piecewise smooth

Lipschitz functions with compact support in X \Xn−2 satisfying the gluing condition for

the the normal derivatives on adjacent (n− 1)-dimensional faces∑∂Pi∩∂Pj 6=∅

∂

∂νj

f = 0 on ∂Pi ∩ ∂Pj (+)

and let ∆X denote the Laplace operator defined on A by ∆Xf(x) =

∑i ∆

Pi(f|Pi)(x)1‖‖Pi

(x).

Then ∆X is essentially self adjoint on L2(X, dm).

Proof. We adopt the proof for smooth case (cf. [Dav89], thm. 5.2.3) with small alterations

which are caused by the exceptional set Xn−2. Since −∆X is non-negative it suffices to

prove that the range of Id−∆X is dense (cf. [RS75], thm. X.26). So suppose u ∈


L2(X, dm) is orthogonal to (Id−∆X )A, i.e.

〈∆X ψ, u〉L2(X,dm) = 〈ψ, u〉L2(X,dm) ∀ψ ∈ A. (2.18)

This means that ∆u = u weakly on each Pi and using elliptic regularity theory we find

that is u is continuous on X and smooth on each Pi up to the boundary except in the

corners Pi ∩Xn−2 which comprises the set where ∂Pi is not smooth. Equation (2.18) also

implies that u satisfies the gluing condition (+) such that we can integrate by parts in

order to obtain

〈∇ψ,∇u〉L2(X,dm) =∑

i

〈∇Piψ,∇Piu〉L2(Pi) = −〈ψ, u〉L2(X,dm) (2.19)

which holds true for all ψ ∈ A and consequently also for all functions ψ ∈ W 1,20 (X \Xn−2)

where we define W 1,20 (X \ Xn−2) as the closure of A with respect to the norm ‖ψ‖2

1,2 =

‖ψ‖2L2(X,dm) +

∑i

∥∥∇Piψ|Pi

∥∥2

L2(Pi). But for η ∈ Lip(X) ∩ Cc(X \ Xn−2) the function ηu

belongs to W 1,20 (X \Xn−2) (note that the condition (+) is void on W 1,2

0 (X \Xn−2)) such

that ψ in (2.19) may be replaced by ηu.

In a second step we would like to get rid of the condition that η vanishes on Xn−2. For

this recall that dimH Xn−2 ≤ n − 2, hence for each compact K ⊂ X we can construct a

sequence of cut-off functions (ρk)k ∈ Lip(X)∩Cc(X \Xn−2) such that ρk → 1 dm-a.e. on

K and ∇ρk → 0 in L2(K, dm) (cf. the very general construction given in [KMS01] which

can be applied in the present context). Consequently, for arbitrary η ∈ Lip(X) ∩ Cc(X)

we may construct a sequence (ρk)k of cut-off functions relative to K = supp(η) as above

in order to obtain a (uniformly bounded) sequence ηk = ρkη ∈ Lip(X) ∩ Cc(X \ Xn−2)

such that ηk → η dm−a.e. and ∇ηk → ∇η in L2(X, dm). Inserting ψ = η2ku in (2.19) and

using Leibniz’ rule for the gradient and Cauchy-Schwarz inequality we obtain∫X

η2k|∇u|2dm = −

∫X

η2ku

2dm− 2

∫X

ηk∇ηku∇udm

≤ 2 ‖ηk|∇u|‖12

L2(X) ‖u|∇ηk|‖12

L2(X) ,

i.e.∫

Xη2

k|∇u|2dm ≤ 4∫

Xu2|∇ηk|2dm which yields

∫Xη2|∇u|2dm ≤ 4

∫Xu2|∇η|2dm <∞

by letting tend k to infinity, i.e. ∇u ∈ L2loc(X, dm). Hence we see that if we set ψ = ρkη

in (2.19)

〈ρk∇(ηu),∇u〉L2(X,dm) + 〈ηu∇ρk,∇u〉L2(X,dm)) = 〈ρkηu, u〉L2(X,dm)

we may in fact pass to the limit for k →∞ in order to prove that

〈∇(ηu),∇u〉L2(X,dm) = 〈ηu, u〉L2(X,dm) ∀η ∈ Lip(X) ∩ Cc(X). (2.20)

The rest of the argument is identical to the proof in [Dav89], which we recapitulate for

the reader’s comfort. Take a smooth function ψ : R → [0, 1] with ψ|[0,1] = 1, ψ|[2,∞) = 0


and let φn : X → [0, 1] be given by

φn(x) = ψ(d(x, y)/n)

where d = dX is the distance function on (X, d) and y ∈ X is some fixed point. Then φn

is a feasible test function for (2.20) and moreover, since |dX(x, y)−dX(z, y)| ≤ dX(x, z) ≤dPi

(z, y) for z, y ∈ Pi and

|∇d(., y)| (x) ≤ lim supz→x

∣∣∣∣dX(x, y)− dX(z, y)

dPi(x, z)

∣∣∣∣ ≤ 1 ∀x ∈ Pi

the bound

‖∇φn‖L∞ = ‖ψ′(d(y, .)/n)∇d(y, .)‖L∞ ≤ ‖ψ′‖∞ /n

obviously holds true. Inserting φ2n for η in (2.20) we get

0 ≥ −〈φ2nu, u〉 = 〈∇(φ2

nu),∇u〉 = 〈φ2n∇u,∇u〉+ 2〈uφn∇φn,∇u〉

which yields ∫X

φ2n|∇u|2dm ≤ 4

∫X

u2|∇φn|2dm ≤ 4

n‖ψ‖∞

∫X

u2dm

and thus finally∇u = 0 a.e. by sending n→∞, which contradicts (2.19) unless u = 0.

The last preparation concerns the construction of the coupling process, where further

singularities of the coupling map Φ(., .) with Φ(x, y) : Rn × Rn → KxX × KyX may

be caused by the existence of non-Euclidean tangent cones Kx if x ∈ Xn−2. However,

choosing beforehand a map Ψ(, ) on X × Xn−2 ∪ Xn−2 × X ∪ (x, x) |x ∈ X with

Ψ(x, y) : Rn × Rn → KxX ×KyX (not necessarily isometric) and depending measurably

on (x, y) we can find a globally defined measurable coupling map Φ(., .) extending Ψ(.)

and satisfying (∗) and (∗∗) on X ×X \ (X ×Xn−2 ∪Xn−2 ×X). This can be proved by

slightly modifying the arguments of lemma 2.1.

Hence we have everything we need to define a sequence of coupled (quasi-)geodesic random

walks on X × X from which we obtain as before the sequences (Ξk,(x,y)

· )k and (Ξk,(x,y)

· )k

by scaling.

Proposition 2.2. For any (x, y) ∈ X×X the sequences (Ξk,(x,y)

· )k and (Ξk,(x,y)

· )k are tight

on DR+(X ×X) and C0(R+, X ×X) respectively. For any subsequence k′ the sequence of

measures (Ξk′,(x,y)

· )∗P on DR+(X ×X) is weakly convergent if and only if (Ξk′,(x,y)

· )∗P is,

in which case the limits coincide. Under any weak limit P∞(x,y) = w- limk′→∞(Ξ

k′,(x,y)

· )∗P

the marginal processes π1. and π2

. are Brownian motions on (X, d) starting in x and y

respectively.


Proof. The tightness assertion and the coincidence of the limits of any jointly converging

subsequences (Ξk′,(x,y)

· )k′ and (Ξk′,(x,y)

· )k′ is proved precisely in the same manner as in

the smooth case. It just remains to identify the limit of the marginal processes. Due

to lemma 2.7 above we know that the generator of (Ec, D(Ec)) is given as the closure of

the Laplacian ∆X =

∑i1‖‖Pi

∆Mi on the set A of piecewise smooth Lipschitz functions

with compact support on X, vanishing on Xn−2 and satisfying (+), i.e. A is a core

for the generator of (Ec, D(Ec)). Now for u ∈ A it is obvious that the sequence Aku

converges locally uniformly and hence also in L2(X, dm) to ∆X u, where the operators Ak

are defined by scaled tangential mean values as in (2.1). Since the (Ak)k are the generators

for the marginals of (Ξk,(x,y)

· )k we may infer from Kurtz’ semigroup approximation theorem

(cf. [EK86], thm. 1.6.1) that also the associated semigroups (P kt )k converge strongly in

L2(X, dm) to the semigroup Pt which is generated by the closure of ∆X , i.e. the semigroup

associated to (Ec, D(Ec)).

Lemma 2.8. Let u ∈ L∞(X, dm) ∩ D(Ec) weakly harmonic on (X, d), i.e. E(u, ξ) = 0

for all ξ ∈ D(Ec), and let P∞(x,y) = w- limk′→∞(Ξ

k′,(x,y)

· )∗P be a weak limit of a subsequence

(Ξk′,(x,y)

· )∗P . Then under P∞(x,y) the processes t → u(ω1

t ) = (u ⊗ 1)(ωt) and t → u(ω2t ) =

(1 ⊗ u)(ωt) are martingales with respect to to the canonical filtration (F t = σπis | s ≤

t, i = 1, 2)t≥0 on CR+(X ×X).

Proof. Note that due to elliptic regularity theory we have that u ∈ C0(X) ∩ C∞(Xn−1)

and that u satisfies the gluing condition (+). Hence we find that Ak(u) → 0 locally

uniformly on X \ Xn−2, where Ak is approximate Laplacian operator (2.1). We would

like to use this property when we pass to the limit for k′ →∞. It remains to justify this

limit. Let us call for short ν = P∞(x,y) = w- limk′→∞(Ξ

k′,(x,y)

· )∗P for a suitable subsequence

k′ and νk = (Ξk,(x,y)

· )∗P .

For ρ > 0 we may find some open neighbourhood Cρ ⊂ X of Xn−2 satisfying

i) Bρ/2(Xn−2) ⊂ Cρ ⊂ Cρ ⊂ Bρ(Xn−2)

ii) ∂Cρ intersects Xn−1 transversally and

iii) ∂Cρ ∩X \Xn−1 is smooth.

Let T iρ = inft ≥ 0|ω1

t ∈ Cρ for i = 1, 2 the hitting time for the marginals of Cρ and

let Di = ω ∈ CR+(X × X) |T iρ is not continuous in ω, then ν(Di) = 0 for i = 1, 2.

This is seen as follows: since the hitting time of a closed set C ⊂ X is lower semi-

continuous on CR+(X) for each ω ∈ Di we necessarily have T iρ(ω) < ∞ and it exists

a sequence ωε → ωi ∈ CR+(X) such that T iρ(ω

i) + δ < lim infε Tiρ(ω

ε) for some δ > 0.

Note that by condition ii) on Cρ the set Xn−1 ∩ ∂Cρ has (Hausdorff-)dimension ≤ n− 2


and hence is polar for Brownian motion on (X, d) and that by proposition 2.2 under ν

the marginal processes are Brownian motions on X. Thus T iρ(ω) < ∞ implies (ν-almost

surely) ωiT i

ρ(ω) ∈ ∂Cρ∩X\Xn−1. But then T iρ(ω

i)+ε < lim infε Tiρ(ω

ε) implies the existence

of some ε0 > 0 such that ωiT i

ρ(ω)+ε′ 6∈ Cρ for all ε′ ≤ ε0. Using the strong Markov property

of the marginal processes under ν and the regularity of ∂Cρ ∩X \Xn−1 we finally deduce

that the set of such paths has indeed vanishing ν-measure.

On account of νk′ ⇒ ν and the ν-almost sure continuity of functional Σρ : DR+(X ×X) → DR+(X × X), (Σρω)(t) = ωt∧T 1

ρ (ω)∧T 2ρ (ω) we find (by thm. 5.1. of [Bil68]) that

(Σρ)∗ νk′ ⇒ νρ := (Σρ)∗ ν for k′ → ∞. Set T ρ = T 1

ρ ∧ T 2ρ , then the Markov property

of Ξk,(x,y)

· and the optional sampling theorem yield that for all t ≥ sl ≥ . . . s1 ≥ 0 and

v, g1, . . . gl ∈ Cb(X ×X)⟨v(ωt)− v(ω0)−

t∫0

Akv(ωs)ds, g1(ωs1) . . . gl(ωsl)

⟩νk

ρ

=

⟨v(ωt∧T ρ

)− v(ω0)−t∧T ρ∫0

Akv(ωs)ds, g1(ωs1∧T ρ) . . . gl(ωsl∧T ρ

)

⟩νk

=

⟨v(ωsl∧T ρ

)− v(ω0)−sl∧T ρ∫0

Akv(ωs)ds, g1(ωs1∧T ρ) . . . gl(ωsl∧T ρ

)

⟩νk

=

⟨v(ωsl

)− v(ω0)−sl∫

0

Akv(ωs)ds, g1(ωs1) . . . gl(ωsl)

⟩νk

ρ

, (2.21)

where Ak is the generator of Ξk,(x,y)

· . If we put v = u⊗ 1 for u as above and use the fact

that

Ak(u⊗ 1) = (Aku)⊗ 1 → 0 uniformly on X \Bρ(Xn−2)

we see that for each t ≥ sl the sequence of functionals (T kt )k defined by

T kt : DR+((X \Bρ(X

n−2))× (X \Bρ(Xn−2)) → R

T kt (ω) = (u⊗ 1)(ωt)− (u⊗ 1)(ω0)−

t∫0

Ak(u⊗ 1)(ωs)dsg1(ωs1) . . . gl(ωsl)

converges uniformly on compacts K ⊂ DR+((X \Bρ/2(Xn−2))× (X \Bρ/2(X

n−2)) to the

functional

Tt(ω) = (u(ω1t )− u(ω1

0))g1(ωs1) . . . gl(ωsl).

defined on DR+(X × X) ⊃ DR+((X \ Bρ/2(Xn−2)) × (X \ Bρ/2(X

n−2)). Hence, due to

νk′ρ ⇒ νρ we may pass to the limit in (2.21) for k′ →∞ giving⟨

(u(ω1t )− u(ω1

0))g1(ωs1) . . . gl(ωsl)⟩

νρ=⟨(u(ω1

sl)− u(ω1

0))g1(ωs1) . . . gl(ωsl)⟩

νρ. (2.22)


Moreover, by definition of νρ (2.22) is equivalent to⟨(u(ω1

t∧T ρ)− u(ω1

0))g1(ωs1∧T ρ) . . . gl(ωsl∧T ρ

)⟩

ν

=⟨(u(ω1

sl∧T ρ)− u(ω1

0))g1(ωs1∧T ρ) . . . gl(ωsl∧T ρ

)⟩

ν. (2.23)

Finally, using again that under ν the marginal processes are Brownian motions on X and

Xn−2 is polar we deduce T ρ ≥ T 1Bρ(Xn−2)∧T 2

Bρ(Xn−2) →∞ for ρ→ 0 ν-almost surely, such

that taking the limit for ρ→ 0 in (2.23) yields⟨(u(ω1

t )− u(ω10))g1(ωs1) . . . gl(ωsl

)⟩

ν=⟨(u(ω1

sl)− u(ω1

0))g1(ωs1) . . . gl(ωsl)⟩

ν,

which amounts to the statement that the process t→ u(ω1t ) is a ((F t), ν)-martingale.

Proposition 2.3. Let (X, d) be an n-dimensional Riemannian polyhedron with lower

sectional curvature bound κ and let for arbitrary x, y ∈ X the measure P∞,(x,y) =

w- liml′→∞

(Ξl′,(x,y)

· )∗P on CR+(X×X) be a weak limit of some suitably chosen subsequence k′.

Then the coupling probability estimate holds true as in the smooth case, i.e.

P∞,(x,y)(TD = ∞) ≤ n− 1

2

√k d(x, y).

Proof. We give only a sketch, because there is essentially nothing new in the arguments.

Using proposition 2.1 we may proceed as in the proof of theorem 2.3 if we restrict of the

discussion onto the set of paths stopped by the stopping time Tρ for ρ > 0. In analogy

to the proof of lemma 2.8 the final step is to send ρ→ 0 which then yields the claim.

Hence with optional sampling from lemma 2.8 and proposition 2.3 we may conclude

Theorem 2.4. Let (X, d) be a d-dimensional generalized locally finite Riemannian poly-

hedron with lower sectional curvature bound κ, then any bounded and weakly harmonic

function u ∈ D(Ec) on (X, d) satisfies the gradient estimate (2.16).

Remark 2.6. It should be noted that the estimate (2.16) is different from Yau’s original

estimate (4.19) because it yields an upper bound for the supremum of the gradient by

the supremum of the function itself, whereas (4.19) is a bound on the supremum of the

logarithmic derivative of a nonnegative harmonic function. In two dimensions our analytic

proof of the stronger estimate (4.19) given in section four covers the case of generalized

two-dimensional Riemannian polyhedra.

Alexandrov Spaces 31

3 Alexandrov Spaces

Basic Concepts

In this section we give a short overview of the most fundamental concepts of Alexandrov

spaces which generalize Riemannian manifolds with lower sectional curvature bound. The

general reference is the paper [BGP92], certain parts of which are thoroughly explained

in [Shi93] and [BBI01].

Most definitions of curvature bounds for metric spaces (X, d) rely on the comparison of

the behaviour of the distance function d on n-tuples of points (xi) ∈ X with the distance

function d on appropriately chosen n-tuples (xi) ∈ Mn,k from the simply connected Rie-

mannian manifold of constant curvature k ∈ R, cf. the concept of K-curvature classes in

[Gro99]. In [BGP92] four essentially equivalent definitions for geodesic metric spaces with

lower curvature bounds are given, among which the four points property is probably the

most general, because it does not even require completeness nor the existence of geodesics

in (X, d).

Definition 3.1 (Metric space with Curvature ≥ k). An arbitrary metric space (X, d)

is said to have curvature bounded from below, i.e. Curv(X) ≥ k, k ∈ R iff every point

x ∈ X admits a neighborhood Ux (with Ux ⊂ Bπ/√

k(x) if k > 0) such that for any

quadruple of points (a; p, q, r) taken from Ux the four-points property holds:

^paq + ^qar + ^rap ≤ 2π. (3.1)

Here ^stv denotes the angle at t of a geodesic triangle ∆(s, t, v) ⊂ M2,k with sidelengths

given by d(s, t) = d(s, t), d(t, v) = d(t, v) and d(v, s) = d(v, s).

By an inner (or path metric) space we mean a metric space with the property that any two

points can be joined by a rectifiable curve with arclength equal to the distance of the given

points, i.e. for all x, y ∈ X there is a (necessarily) continuous curve γxy : [0, d(x, y)] → X

such that L(γxy) = d(x, y), where the length of a curve c : [a, b] → X is defined by

L(c) = sup∑

a=t0≤t1≤···≤tK=b

d(c(ti+1), c(ti))

with the supremum being taken over all finite partitions a = t0 ≤ t1 ≤ · · · ≤ tK = b.

For locally compact and complete metric spaces this is equivalent to the existence of a

midpoint, i.e. for all x, y ∈ X there is a z ∈ X such that

d(x, z) = d(y, z) =1

2d(x, y).

A curve joining two points x and y as above is called a geodesic segment, any curve being

locally a geodesic segment is called geodesic (arc). We may assume without restriction

32 Alexandrov Spaces

that any geodesic is parametrized according to arclength. Finally a geodesic triangle in

an inner metric space is defined by its three vertices and three geodesic segments joining

each pair of them. In the sequel we are going to deal with locally compact inner metric

spaces only.

Definition 3.2 (Alexandrov Space). A locally compact and complete inner metric

space (X, d) with Curv(X) ≥ k is called Alexandrov space with curvature bounded below

by k.

Proposition 3.1 (Alexandrov Convexity, local version). For a locally compact inner

metric space the condition Curv(X) ≥ k is equivalent to the Alexandrov convexity of

geodesic hinges: for each x ∈ X there is an ε > 0 such that for any pair of geodesics γ, η

with γ(0) = η(0) = x the function

θ(s, t) = ^(γ(s)xη(t)) is non-increasing for s, t ≤ ε.

This formulation is very important because it permits the introduction of the following

crucial concepts.

Definition 3.3 (Angles and Tangent Cones). Let γ, η be two geodesics emanating

from x ∈ X as above then

^(γ, η) := lims,t→0

θ(s, t) =: d^(γ, η)

defines the angle (or angular distance) between η and γ. The space of directions (Σx, d^)

is the (closure of the) set of equivalence classes of all geodesics emanating from x with

respect to the angular distance d^

Σx =(γxy|y ∈ X/∼d^

, d^

)∼.

The tangent cone (Kx, dx) is the topological cone (Σx×R+)/(Σx×0) over Σx equipped

with the metric induced on R+ × Σx by the Euclidean cosine law

d2x[(α, s), (β, t)] = s2 + t2 − 2st cos d^(α, β).

Remark 3.1. In fact, in the definition above it would be better to talk of the angular

distance between γ and η only, because ^(γ, η) as defined above yields the shorter of the

two angular rotations from γ to η and from η to γ respectively, i.e. it does not distinguish

between ’inner’ or ’outer’ angle.

Generally, for a geodesic triangle ∆(p, q, r) ⊂ X, which is defined by its vertices and

arbitrary but fixed geodesic segments pq, qr and rp ⊂ X, let ∆(p, q, r) = ∆(p, q, r) denote

the (up to congruence) uniquely defined geodesic triangle in M2,k such that d(p, q) =

d(p, q), d(q, r) = d(q, r) and d(r, p) = d(r, p), which will be called comparison triangle for


∆(p, q, r). If k ≤ 0 one can always find such a triangle, in the case k > 0 one has to

impose for its circumference d(p, q) + d(r, q) + d(q, p) ≤ 2π/√k. Having introduced the

notion of angles between geodesics one can now state two more equivalent properties (cf.

[BGP92]):

Proposition 3.2. For a locally compact inner metric space the following properties are

equivalent:

i) Curv(X) ≥ k (in the sense of definition 3.1)

ii) for every geodesic triangle ∆(p, q, r) ⊂ Ux the angles are bounded above by the

corresponding angles of the comparison triangle ∆(p, q, r) ⊂ M2,k

iii) for every geodesic triangle ∆(p, q, r) ⊂ Ux and a corresponding comparison triangle

∆(p, q, r) ⊂ M2,k the inequality

d(p,mqr) ≥ d(p,mqr)

holds, where mqr ∈ pq and mqr ∈ qr are the midpoints of the geodesics rq and rq in

X and M2,k respectively.

The last statement means that every geodesic triangle ∆ ⊂ Ux ⊂ X in such a neigh-

borhood of any given point in x ∈ X is more convex than the corresponding comparison

triangle ∆ ⊂ M2,k, where the convexity of a triangle is measured by the distances from its

vertices to interior points on the opposing side. Finally, all previous essentially equivalent

local conditions can be turned into global ones ([BGP92], thm. 3.2.):

Theorem 3.1 (Globalization Theorem). If (X, d) is a complete space with Curv(X) ≥k, then inequality (3.1) remains true for any geodesic triangle in (X, d).

Remark 3.2. In the case k > 0 the assumption about the circumference of triangles is

superfluous since it will be automatically satisfied. Also the diameter of X is bounded

from above by π/√k.

Examples 3.1. ([BGP92, Shi93])

i) Riemannian manifolds with sectional curvature bounded from below and quotients

M/Γ of such Riemannian manifolds by groups Γ acting isometrically (not necessarily

free or discrete)

ii) simplicial n-dimensional Riemannian complexes (obtained from gluing together Rie-

mannian simplexes of constant curvature k) which satisfy the 2π-gluing-condition

along the faces of codimension 2

34 Alexandrov Spaces

iii) boundaries of convex subsets in Riemannian manifolds with lower sectional curva-

ture bound, and as a special case surfaces of revolution ⊂ R3 obtained from graphs

of convex functions

iv) spherical suspensions or cones of Alexandrov spaces, for instance C(RP n), the Eu-

clidean cone over the real projective space of dimension n

v) spaces obtained by gluing two Alexandrov spaces along their boundaries if the

boundaries are intrinsically isometric ([Pet97])

vi) Hausdorff-limits of Riemannian manifolds with uniform sectional curvature bound.

- For this recall the definition of the Hausdorff-distance of two subsets A, B ⊂ (X, d)

of a metric space

dXH(A,B) = infε > 0 |A ⊂ Uε(B), B ⊂ Uε(A).

Now the Hausdorff-distance§ of two metric spaces (A, dA), (B, dB) is defined by

dH(A,B) = inf dXH(f(A), g(B))

where the infimum is taken over all metric spaces (X, d) and all isometric embeddings

f : A → X and g : B → X. Then dH induces a complete metric between compact

metric spaces (cf. [Gro99]), i.e. the space Xc of (equivalence classes of) compact

metric spaces together with the function dH forms a complete (and contractible) not

locally compact metric space. By Gromov’s compactness theorem for each choice of

n ∈ N, κ ∈ R, D ∈ R+ the set M(n, κ,D) of n-dimensional Riemannian manifolds

with lower Ricci curvature bound (n−1)κ and diameter less than D is precompact in

(Xc, dH). Moreover, if one assumes also the sectional curvature to be bounded below

by k ∈ R then any Hausdorff limit of a converging sequence will be an Alexandrov

space with curvature bounded below by k (cf. [BBI01]).

Dimension and Regularity

In contrast to singular spaces with upper curvature bounds Alexandrov spaces exhibit a

great amount of regularity and even carry a natural weak Riemannian structure. Firstly,

each Alexandrov space of finite rough dimension¶ has in fact integer Hausdorff dimension,

which then coincides with its topological dimension. The proof of this fact is based upon

§The Lipschitz distance dL(A,B) between two metric spaces (A, dA) and (B, dB) is defined bydL(A,B) = inf| log dilf | + | log dilf−1|

∣∣f : A → B bi-Lip homeom.. For compact spaces the conver-gence with respect to the Hausdorff-distance is essentially equivalent to the Lipschitz convergence ofcorresponding ε-nets for arbitrary ε > 0, cf. [Gro99, Pet86].

¶The rough dimension of a bounded subset Z ⊂ (X, d) is defined as dimr(Z) = infα >

0 | limε→0 βZ(ε)εα = 0, where βZ(ε) is maximal number of points ai ∈ Z with d(xi, xj) ≥ ε if i 6= j .


the local constancy of the rough dimension in X and the existence of a dense open set

of manifold points p, which are characterized by existence of a maximal system of almost

orthogonal almost geodesic curves passing through p (strainer at p), which in turn can

be used for the construction of bi-Lipschitz homeomorphisms of a certain neighborhood

Up ⊂ X onto an open subset Rn (see the appendix B for a short sketch of the ideas).

In particular each finite dimensional Alexandrov space is locally compact and supports

a natural nontrivial measure mX , which is its n-dimensional Hausdorff measure when

dim(X) = n ∈ N. Throughout the rest of this work we are going to deal with finite

dimensional Alexandrov spaces only. For each δ > 0 the system of natural coordinate

functions

φai : X ⊃ Vp → Rn q → (d(a1, q), . . . , d(an, q))

which are induced from the (n, δ)-strainers‖ (a−i, a+i) | i = 1, . . . , n at p ∈ Xδ = X \ Sδ

yields a (locally bi-Lipschitz) topological atlas for Xδ for sufficiently small δ ([BGP92]).

Definition 3.4. Let Sδ = p ∈ X |@ (n, δ)-strainer at p. Then the set SX =⋃

δ>0 Sδ =

limδ→0 Sδ is called the (metrically) singular set.

SX is equivalently characterized by

SX = x ∈ X |Σp 6' Sn−1,

which follows from the compactness of Σp and a suitable sphere theorem (cf. thm. 8.9 in

[Shi93]) and it can be shown that dimH(SX) ≤ n− 1 ([BGP92, OS94]), but except being

the countable union of closed sets no further topological information on SX is obtained.

Conversely, X\SX =⋂

0<δ1Xδ withXδ = x ∈ X | rad(Σx) > π−δ∗∗ is the intersection

of Lipschitz manifolds which are open in X. However, X \ SX need not be a manifold in

general but it still carries a so called weak C1-Riemannian structure††, which is revealed

by a careful analysis of the regularity properties of the natural coordinate functions.

Theorem 3.2 (Weak C1-Riemannian Structure ([OS94])). For properly chosen

Uφaithe system of maps

φai : X ⊃ Uφai

→ Rn | (a−i, a+i) is a (n, δ)-strainer at p, p ∈X\SX , δ ≤ 1/2n

gives rise to a weak C1-Riemannian structure on X\SX , whose induced

intrinsic distance coincides with the initial distance d on X \ SX .

In section 5.1 it will be shown that the metrically singular points do not comprise all

points in which the local behaviour of X deviates considerably from that of a smooth

Riemannian manifold.

‖cf. appendix B for the precise definition of a (n, δ)-strainer at p ∈ X∗∗The radius rad(A) of a subset A in a metric space (X, d) is defined by rad(A) := inf

ξ∈Asupη∈A d(ξ, η).

††cf. appendix B for a definition of a weak Riemannian structure

36 Gradient Estimates on Alexandrov Surfaces


In this section we give an analytic proof of gradient estimates for harmonic functions on

two-dimensional Alexandrov spaces with Lipschitz continuous metric tensor (gij). Since

the paper [CH98] contains a proof of gradient estimates for Riemannian manifolds with

Lipschitz metric tensor satisfying a distributional lower curvature bound it remains for

us to derive this property for Alexandrov surfaces out of the geometric curvature bounds

given in terms of geodesic triangles‡‡. By our result we can enlarge significantly the set

of examples of non-smooth spaces given in [CH98] which admit Yau’s gradient estimate.

4.1 An Integral Gauss-Bonnet-Formula

Thus the main task is to overcome the technical difficulties arising from the low regularity

of the metric tensor, the main step being an extension of the Gauss-Bonnet theorem to a

non-smooth situation. Here we rely on the existence and semi-boundedness of the excess

measure eX on an arbitrary Alexandrov space as introduced and investigated in the paper

[Mac98], see also section 6.2 for a short exposition. Let h(x) denote the open Euclidean

square in R2 of sidelength h > 0 with lower left corner x and assume that h(x) is

contained in the image Ω = ψ(U), U ⊂ X of a natural local chart ψ of (X, d) and let

ex(h(x)) denote the excess measure of the set ψ−1(h(x)) ⊂ X. Then we obtain the

following

Proposition 4.1 (Gauss-Bonnet). Let (X, d) be a two-dimensional Alexandrov space

with Curv(X) ≥ κ ∈ R and discrete metrically singular set SX such that the system

of natural coordinates with the associated metric tensors (gij) yield a C1,1 Riemannian

structure for X \ SX , i.e. the chart transition maps are C1,1loc and the metric tensors

(gij) are locally uniformly Lipschitz continuous. Let Ω = φai(U) ⊂ R2 be the image

of U ⊂ X \ SX under φai and let gai = (gij) : Ω → R2×2, gij ∈ Liploc(Ω), be the

Riemannian tensor on U with respect to φ. Then for all h > 0 and almost all x ∈ Ω with

dist(x, ∂Ω) > h the set h(x) is eX-measurable and for any ξ ∈ C0c (Ω) and sufficiently

small h > 0 the following identity applies:∫Ω

eX(h(x))ξ(x)dx =

∫Ω

ξ(x)( 4∑

i=1

înti (h(x))− 2π

)dx

−∫Ω

h∫0

[ξ(x− te2)

√det gg22

(x)Γ122(x)− ξ(x− he2 − te1)

√det gg11

(x)Γ211(x)

]dt dx (4.1)

+∫Ω

h∫0

[ξ(x− (h− t)e2 − he1)

√det gg22

(x)Γ122(x) −ξ(x− (h− t)e1)

√det gg11

(x)Γ211(x)

]dt dx .

‡‡Compare the works by I. Nikolaev (cf. survey in [BNR93]) who proved that metric space spaces withtwo-sided curvature bounds can be approximated in Gromov-Hausdorff distance by smooth Riemannianmanifolds. Here we have only lower curvature bounds but impose a priori a certain regularity on (X, d).

Gradient Estimates on Alexandrov Surfaces 37

Remark 4.1. In the case (gij) ∈ BVloc(Ω)∩C0(Ω) - at least on a formal level - it would

be perfectly possible to replace the measures Γijj(x)dx in the formula above by the finite

Radon measures Γijj(dx), but unfortunately we were not able to prove (4.1) under these

weaker assumptions. Because of this reason we state our main theorem of this section

under the assumption that (gij) ∈ Liploc(Ω).

For the proof of proposition 4.1 some preparation is needed. At first we take a look at the

smooth situation. Suppose (M, g) is a smooth (two-dimensional) Riemannian manifold

and let Ω ⊂ R2 be contained in the image of some chart ψ : M ⊃ U → Ω ⊂ R2 with

associated metric tensor gij. For every x ∈ Ω and v ∈ R2, s > 0 we study the behaviour

of the geodesic γvx,s = γx,x+sv on the manifold M connecting the points that correspond

to the Euclidean points x and x+ sv ∈ Ω for small s. Throughout this section we assume

that every geodesic between fixed endpoints p, q ∈M is parameterized on the unit interval

and we identify γ with its image under ψ as long as it is entirely contained in U . Then

we obtain the following

Lemma 4.1. In the smooth case for s tending to zero the asypmtotic formula

γvx,s(0) = sv − s2

2Γx(v, v) + o(s2) (4.2)

holds true locally uniformly with respect to x ∈ Ω, v ∈ R2.

Proof. The assertion is a consequence of the geodesic equation. By definition of the

geodesic γvx,s

γvx,s(ρ)− sv = γv

x,s(ρ)− (γvx,s(1)− γv

x,s(0)) =

1∫0

[γvx,s(ρ)− γv

x,s(τ)]dτ

=

1∫0

ρ∫τ

γvx,s(σ)dσdτ =

1∫0

ρ∫τ

Γγvx,s(σ)(γ

vx,s(σ), γv

x,s(σ))dσdτ (4.3)

≤ Cs2

1∫0

∣∣∣∣∣∣ρ∫

τ

∥∥∥Γγvx,s(σ)

∥∥∥ dσ∣∣∣∣∣∣ dτ ≤ Cs2 sup

σ∈[0,1]

∥∥∥Γγvx,s(σ)

∥∥∥ 1∫0

|ρ− τ |dτ = O(s2) (4.4)

where we have put Γz : R2 × R2 → R2, Γz(ξ, η) = Γkij(z)ξiηjek, z ∈ Ω and O(s) is a ”big

O” Landau-function. In (4.3) we used the fact that∥∥γvx,s

∥∥gγv

x,s

= const = d(x, s+ sv) ≤ Cgs

locally uniformly w.r.t. x and v. Hence γvx,s(ρ) = sv + O(s2) for all ρ ∈ [0, 1]. Plugging

this back into the second integral integral of (4.3) for the case ρ = 0 gives

γvx,s(0)− sv = −s

2

2Γx(v, v) + s2

1∫0

τ∫0

[Γx − Γγv

x,s(σ)

](v, v)dσdτ + o(s2) (4.5)


From the continuity of Γ(.) it follows that1∫0

∫ τ

0

[Γx − Γγv

x,s(σ)

](v, v)dσdτ = ϑ(s) and since

all the error estimates above hold true locally uniformly with respect to x and v the claim

of the lemma is established.

Our next goal is the derivation of (4.2) in the given non-smooth situation. Here the main

device is the approximation of (gij) by smooth metrics (gεij = φε ∗ gij), where φε is a

sequence of smooth symmetric Dirac functions.

Lemma 4.2. The sequence (gεij) ∈ R2×2

sym has the following properties

(gεij)

ε→0−→ (gij) locally uniformly on Ω and

gεij,k

ε→0−→ Dxkgij strongly in Lp

loc(Ω) for all p ∈ (1,∞)∥∥gεij,k

∥∥C0(K)

≤ [gij]Lip(K)for all K ⊂ Ω compact.

With (gεij) given as above and v ∈ R2 define furthermore

γv,εx,s(δ) =

1

mε(Bs3(x+ sv))

∫B1(0)⊂R2

γεx,x+s(v+s2θ)(δ)

√det gε

ij(x+ s(v + s2θ))dθ

γvx,s(δ) =

1

m(Bs3(x+ sv))

∫B1(0)⊂R2

γx,x+s(v+s2θ)(δ)√

det gij(x+ s(v + s2θ))dθ

(4.6)

where γεx,x+s(v+s2θ) : [0, 1] → Ω and γx,x+s(v+s2θ) : [0, 1] → Ω denote geodesic segments

with respect to (gεij) and (gij) respectively connecting the point x with x + s(v + s2θ),

where θ ∈ B1(0) and mε(Bs3(x+ sv)) =∫

Bs3 (x+sv)(det gε

ij(y))12dy, m(Bs3(x+ sv) =∫

Bs3 (x+sv))(det gij(y))

12dy. This complicated choice of γv

x,s(δ) and γv,εx,s(δ) is motivated by

the following property.

Lemma 4.3. For all x ∈ Ω and δ ∈ [0, 1] limε→0 γv,εx,s(δ) = γv

x,s(δ).

Proof. Since gεij → gij uniformly, we obtain that for each pair of points x, y in Ω suffi-

ciently close the sequence of geodesics γεxy is uniformly bounded in W 1,∞(R+, U) = γ :

R+ → U | dx-ess sup√γigij(γ)γj < ∞ which implies in particular the precompactness

of this sequence w.r.t. (locally) uniform convergence. Since the arclength functional

L(gij) is lower semicontinuous w.r.t. to weak convergence in W 1,2([0, 1], U) = γ : R+ →U | ‖

√γigij(γ)γj‖L2(R+,dx) < ∞ any limit function of γε

xy will be geodesic segment from

x to y in X. From the fact that the cut locus Cx = y ∈ X | γxy not unique of every

point x ∈ X has zero Hausdorff measure in X ([OS94], prop. 3.1) we may conclude the

assertion of the lemma.


Remark 4.2. The estimate (4.4) prevails for the (gij)-geodesic segments γvx,s in the sense

that

∀K ⊂ Ω ∃C ∈ R+ : m-ess supθ∈Bs3 (x+sv)

‖γx,θ(0)− sv‖ ≤ Cs2 ∀x ∈ K, (4.7)

which may be seen as follows: using the analogue of (4.4) for the (gεij)-geodesics γε

x,θ, θ ∈Bs3(x + sv), we obtain the corresponding estimate (4.7ε) for the approximating metrics

(gεij), where the constant C on the right hand side can be chosen uniform with respect

to ε. (This is a simple consequence of the uniform bound on ‖Γε(., .)‖). Hence we may

pass to the limit in (4.7ε) for ε → 0 provided that γεx,θ(0) → γx,θ(0). However, by the

geodesic equation and by the uniform boundedness of ‖Γε(., .)‖ it is evident that the

Lipschitz norm of γεx,θ(.) is uniformly bounded, which implies relative compactness of the

family (γεx,θ(.))ε≥0 in C1([0, 1],R2). Together with the m-almost sure uniqueness of the

(gij)−geodesic γx,θ with respect to the endpoint θ we may conclude γεx,θ(0) → γx,θ(0) for

m−a.e. θ such that we can in fact pass to the limit for m-almost all θ ∈ Bs3(x + sv)

which yields the assertion. - From the same compactness and uniqueness arguments we

may conclude also that the function Ω 3 x→ γv

x,s(0) with

γv

x,s(0) := limε→0

γε,v

x,s(0) =∫–

Bs3 (x+sv)

γx,θ(0)m(dθ) = limδ→0

1

δ

[γv

x,s(δ)− x]

= γv

x,s(0)

is well defined and locally uniformly bounded , i.e. γv

.,s(0) ∈ L∞loc(Ω).

Now we can formulate the key lemma for the proof of proposition 4.2.

Lemma 4.4. Under the conditions of proposition 4.1 formula (4.2) remains true for

γv

x,s(0) in integrated sense, i.e. for all ξ ∈ C0c (Ω)

lims→0

1

s2

∫Ω

ξ(x)[γ

v

x,s(0)− sv]dx = −1

2

∫Ω

ξ(x)Γx(v, v)dx (4.8)

Proof. For smooth (gij) from lemma 4.1 we obtain (4.8) because the error term in (4.2)

is estimated by o(s2) locally uniformly w.r.t. x and v and thus

γv

x,s(0) =1

m(Bs2(x+ sv))

∫B1(0)

γx,x+s(v+s2θ)(0)√


=1

m(Bs2(x+ sv))

∫B1(0)

s(v + s2θ)− s2

2Γx(v + s2θ, v + s2θ) + o(s2)

×√


= sv − s2

2Γx(v, v) + o(s2)

For general (gij) ∈ Liploc(Ω) we use the approximation by smooth (gεij) furnished by

lemma 4.2. Hence for all ε > 0 we obtain equation (4.8) with the quantities γv

x,s and gij


replaced by γv,ε

x,s and gijε, which we denote (4.8ε). In order to pass to the limit in (4.8ε)

for ε tending to zero it is sufficient to show that

i) for s fixed∫Ω

ξ(x)γv

x,s(0)dx = limε→0

∫Ω

ξ(x)γv,ε

x,s(0)dx and

ii) the convergence in (4.8ε) for s tending to zero is uniform with respect to ε.

i) From lemma 4.3 we know that∫Ω

ξ(x)[γv,ε

x,s(δ)− x]dx

ε→0−→∫Ω

ξ(x)[γv

x,s(δ)− x]dx ∀ ξ ∈ C0

c (Ω), s ≥ 0, δ ≥ 0.

Hence, if we can show that the limit

limδ→0

1

δ

∫Ω

ξ(x)[γv,ε

x,s(δ)− x]dx =

∫Ω

ξ(x)γv,ε

x,s(0)dx (4.9)

is uniform w.r.t. ε then i) is proved. In fact, for any compact K ⊂ Ω and x ∈ K∣∣1δ[γε

x,s(δ)− x]−γε

x,s(0)∣∣

=

∣∣∣∣∣∣∣1

δ

δ∫0

σ∫0

1

mε(Bs3(x+ sv))

∫Bs3 (x+sv)

γεx,θ(τ)m

ε(dθ)dτdσ

∣∣∣∣∣∣∣≤ 1

δ

δ∫0

σ∫0

1

mε(Bs3(x+ sv))

∫Bs3 (x+sv)

∥∥∥Γεγε

x,θ(τ)

∥∥∥R2×R2→R2

mε(dθ)dτdσ

≤ 1

2δ sup

x∈B2s(K)

‖Γεx‖ ≤ CK,sδ (4.10)

by the uniform boundedness of supx∈K

‖Γεx‖R2×R2→R2 for all K ⊂ Ω compact. Obviously this

implies (4.9).

ii) By definition of γε,vx,s

γε,v

x,s(0)− sv =∫–

Bs3 (x+sv)

(γε

x,θ(0)− [γεx,θ(1)− γε

x,θ(0)])mε(dθ)

+∫–

Bs3 (x+sv)

(sv + x− θ)mε(dθ).

The uniform boundedness of supx∈K

‖Γεx‖R2×R2→R2 for all K ⊂ Ω compact implies via (4.3)

for γε,vx,s that γε,v

x,s(ρ) = sv + O(s2)∀ ρ ∈ [0, 1], x ∈ K, v ∈ K ′ uniformly w.r.t. ε > 0, such

that we can repeat the steps leading to (4.5) in a similar manner which yields

γε,vx,s(0)− sv = −s

2

2Γε

x(v, v) + s2

1∫0

τ∫0

∫–

Bs3 (x+sv)

[1

2Γε

x − Γεγε

x,θ(σ)

](v, v)mε(dθ)dσdτ + o(s2).


If we multiply this equation with a compactly supported test function ξ and integrate

with respect to x we obtain∣∣∣∣∣∣ 1

s2

∫Ω

ξ(x)[γε,vx,s(0)− sv]dx+

1

2

∫Ω

ξ(x)Γεx(v, v)dx

∣∣∣∣∣∣ ≤ 1

2‖ξ‖L2 ‖Γε

. (v, v)− Γε,s. (v, v)‖L2+ϑ(s)

with

Γε,sx (v, v) = 2

1∫0

τ∫0

∫–

Bs3 (x+sv)

Γεγε

x,θ(σ)(v, v)mε(dθ)dσdτ

which converges by a general property of Lp-functions to Γ.(v, v) in Lp for all p ∈ (1,∞)

if ε and δ tend to zero. This proves ii) and hence the lemma.

The subsequent considerations require a clarification of the meaning of angles between

geodesics emanating from the same point: for each x ∈ Ω ⊂ X we have a natural

orientation on the tangent cone Kx as induced from the chart φai and the orientation

of R2. Hence we may talk about rotations of Kx in positive or negative direction and

consequently if Kx is non-singular for given θ ∈ Σx any η ∈ Σx is uniquely determined by

cos ^(θ, η) and the orientation of the basis η, θ in Kx. Moreover, the distinction between

^(θ, η) and ^(η, θ) makes sense and which are related by the formula ^(θ, η) + ^(η, θ) =

L(Σx), where L(Σx) denotes the length of the space of directions over x. If Kx is regular,

equivalently if L(Σx) = 2π, and γ and η are geodesic segments parameterized by arclength

emanating from x then we set

^(γ, η) = sign(γ, η)d^(γ, η) mod 2π,

where

sign(γ, η) =

+1 if γ, η pos. orientated in Kx

−1 if γ, η neg. orientated in Kx.

If x ∈ X is contained in an open orientable domain O ⊂ X, i.e. if the chart transitions

φai φ−1bi preserve the orientation of Ky for y ∈ O then this definition of the angle

between two geodesic segments is obviously independent of the specific choice of φai.

Moreover, if the images of γ and η under φai are C1-curves in Ω, the properties of the

weak Riemannian structure imply

d^gx(γ, η) = arccos(〈γ, η〉gx)

and analogously we define the angle ^gx between two arbitrary C1-curves γ, η in Ω ⊂ R2

starting from x ∈ Ω measured w.r.t. (gij) by the same formula as above

^gx(γ, η) = sign(γ, η)d^gx(γ, η). (4.11)


Corollary 4.1. Under the conditions of proposition 4.1 for all ξ ∈ C0c (Ω)

lims→0

1

s

∫Ω

∫–

Bs3 (x+sv)

ξ(x)[^gx(γxθ, v

⊥x )− π/2

]m(dθ)dx =

1

2

∫Ω

ξ(x)1

‖v‖gx

〈Γx(v, v), v⊥x 〉gxdx

= lims→0

1

s

∫Ω

∫–

Bs3 (x−sv)

ξ(x)[^gx(v

⊥x , γθx)− π/2

]m(dθ)dx (4.12)

where the vectors v⊥x and v⊥θ are are obtained from v by a positive rotation of v about the

angle π/2 with respect to the angular distance d^gxand d^gθ

respectively.

Proof. We show the first identity in (4.12), the proof of the second is similar. By (4.7)

we find∣∣∣∣sin ^gx(γxθ, v)

2

∣∣∣∣ =1

2

∥∥∥∥∥ γxθ

‖γxθ‖gx

− v

‖v‖gx

∥∥∥∥∥gx

≤ ‖γxθ − sv‖ ‖sv‖ | ‖sv‖ − ‖γxθ‖ |‖sv‖ ‖γxθ‖

= O(s)

(4.13)

for m-a.e. θ ∈ Bs3(x + sv) which implies via the Taylor-expansion of arcsin about zero,

i.e. ^gx(γxθ, v) = sin ^gx(γxθ, v) + o(s2), that ^gx(γxθ, v) = O(s). From this and from the

triangle inequality for angular distances we see that |^gx(γxθ, v⊥x )−π/2| = O(s) and thus

^gx(γxθ, v⊥x )− π/2 = sin

(^gx(γxθ, v

⊥x )− π/2

)+ o(s2) = − cos ^gx(γxθ, v

⊥x ) + o(s2)

for m-a.e. θ ∈ Bs3(x + sv) locally uniformly w.r.t. x ∈ Ω. Hence we may replace

^gx(γxθ, v⊥x )− π/2 by

− cos ^gx(γxθ, v⊥x ) = −〈 γxθ

‖γxθ‖gx

, v⊥x 〉gx = −〈 γxθ − sv

‖γxθ‖gx

, v⊥x 〉gx .

in the first integral in (4.12), which we rewrite

−1

s

∫Ω

∫–

Bs3 (x+sv)

ξ(x)〈 γxθ − sv

‖sv‖gx

, v⊥x 〉gxm(dθ)

− 1

s

∫Ω

∫–

Bs3 (x+sv)

ξ(x)〈(γxθ − sv)(‖sv‖gx

− ‖γxθ‖gx)

‖γxθ‖gx‖sv‖gx

, v⊥x 〉gxm(dθ).

(4.14)

Furthermore, from (4.10) and the fact that γεxθ → γxθ for m-a.e. θ ∈ Bs3(x+sv) it follows

that (4.4) remains true for m-a.e. θ ∈ Bs3(x+ sv) locally uniformly w.r.t. x and hence∥∥∥∥∥(γxθ − sv)(‖sv‖gx− ‖γxθ‖gx

)

‖γxθ‖gx‖sv‖gx

∥∥∥∥∥ ≤ O(s2)O(s2)

‖v‖ s(1 +O(s2))for m-a.e. θ ∈ Bs3(x+ sv)

which implies that the second integral in (4.14) tends to zero for s → 0. Since all the

functions depending on x except γxθ in the first integral of (4.14) are continuous and are

cut-of by ξ equality (4.8) proves the claim.


Remark 4.3. The proofs of lemma 4.4 and corollary 4.1 show that if we replace ξ(x) in

(4.12) by ξ(x+ λw) with λ ∈ R and w ∈ R2 then the convergence is locally uniform with

respect to λ.

Proof of proposition 4.1. For k ∈ N and x ∈ Ω divide each side of the boundary ∂(h(x))

=⋃4

i=1 ∂i(h(x)) into 2k subintervals of length s = sk = h

2k. Let (vi

h(x))i=1,...,4, v1

h(x) = x

denote the four vertices of the square h(x) (numbered in clockwise direction) and

(σ1j )

i=1,...,4j=1,...,2k−1 the interior partitioning points on the sides of ∂(h(x)). Further set σi

0 = vi

and σi2k = vi+1 where the upper index i is always taken modulo 4. Then for each odd

interior partitioning point σi2l+1 (l = 0, . . . , k − 1) we choose some θi

2l+1 in the Euclidean

ball Bs3(σi2l+1) and (w.r.t. (gij)) geodesic segments γσi

2lθi2l+1

and γθi2l+1σi

2(l+1)connecting

θi2l+1 with its neighboring even partitioning points σi

2l and σi2(l+1) respectively. The con-

catenation of these segments defines the boundary of a polygonal domain h(x)[k, (θji )]

depending on the choice of the (θij). By the local equivalence of the Euclidean and

the (w.r.t. (gij)) Riemannian distances on Ω the topological type of h(x)[k, (θji )] in

R2 and in the Alexandrov space (X, d) are the same so that if the geodesic segments

γσi2lθ

i2l+1

and γθi2l+1σi

2(l+1)are close to the corresponding Euclidean segments one may con-

clude that the domain h(x)[k, (θji )] is simply connected in (X, d) and orientable, i.e. for

the Euler-Poincare characteristic of h(x)[k, (θji )] in X we obtain χ

X(h(x)[k, (θ

ji )]) = 1.

Consequently, the combinatorial Gauss-Bonnet theorem (X, d) (cf. [Mac98]) applied to

the domain h(x)[k, (θji )] yields

eX(h(x)[k, (θji )]) = 2π −

∑i=1,...,4

l=0,...,k−1

(π − înt(γσi

2lθi2l+1

, γθi2l+1σi

2(l+1)))

−∑

i=1,...,4l=1,...,k−1

(π − înt(γθi

2l−1σi2l, γσi

2lθi2l+1

))−∑

i=1,...,4

(π − înt(γθi−1

2k−1vi , γviθi1)).

(4.15)

The idea is now, of course, to multiply this equation by a test function ξ(x), integrate with

respect to x and (θij) and let tend k tend to infinity. Taking the mean in (4.15) over all

(θij) ranging over the Euclidean ball Bs3(σi

2l+1) with respect to the Riemannian measure

m induced from (gij) and integrating this with respect to dx against a test function gives

the identity

∫Ω

ξ(x)∫–

Bs3 (σ11(x))

· · ·∫–

Bs3 (σ42k−1(x))

eX(h(x)[k, (θji )(x)])m(dθ1

1) · · ·m(dθ42k−1)dx = Ik(h) (4.16)


with

Ik(h) =

∫Ω

ξ(x)

4∑i=1

∫–

Bs3 (σi−12k−1)

∫–

Bs3 (σi1)

înt(γθ1vi , γviθ2)m(dθ1)m(dθ2)− 2π

dx−∫Ω

ξ(x)4∑

i=1

k−1∑l=0

∫–

Bs3 (σi2l+1)

(π − înt(γσi

2lθ, γθσi

2l+2))m(dθ)dx

−∫Ω

ξ(x)4∑

i=1

k−1∑l=0

∫–

Bs3 (σi2l−1)

∫–

Bs3 (σi2l+1)

(π − înt(γθ1σi

2l, γσi

lθ2))m(dθ1)m(dθ2)dx.

Let us identify the limit limk→∞ Ik(h) first. We rewrite Ik(h) as∫Ω

ξ(x)

4∑i=1

∫–

Bs3 (σi−12k−1)

∫–

Bs3 (σi1)

înt(γθ1vi , γviθ2)m(dθ1)m(dθ2)− 2π

dx−

4∑i=1

h∫0

∫Ω

ξ(x)f ik(x, t)dxdt

(4.17)

with f ik : Ω× [0, h] → R defined for i = 1, . . . , 4 by

f ik(x, t) =

1sk

∫–

Bs3 (σi2l+1)

(π − înt(γσi

2lθ, γθσi

2l+2))m(dθ)

if t ∈ ( h2k

(2l + 1)− sk

2, h

2k(2l + 1) + sk

2], l = 0, . . . , k − 1

1sk

∫–

Bs3 (σi2l−1)

∫–

Bs3 (σi2l+1)

(π − înt(γθ1σi

2l, γσi

2lθ2))m(dθ1)m(dθ2)

if t ∈ ( h2k

(2l)− sk

2, h

2k(2l) + sk

2], l = 1, . . . , k − 1

0 if t ∈ [0, sk

2] ∪ (h− sk

2, h].

Estimate (4.7) and a discussion similar to the proof of corollary 4.1 show that the first

integral in (4.17) converges to∫

Ωξ(x)

(∑4i=1 înt

i (h(x))− 2π)m(dx). (This could also be

deduced directly from (4.12) and the decomposition of angles in Euclidean tangent cones,

compare (4.18) below). So it remains to prove the convergence of∫ h

0

∫Ωξ(x)f i

k(x, t)m(dx)dt.

Let us consider f 1k first. Note that relative to the orientation on R2 for the interior angles

above we have

înt(γσi2lθ, γθσi

2l+2) = ^gθ

(γσi2lθ, γθσi

2l+2) and înt(γθ1σi

2l, γσi

2lθ2) = ^g

σi2l

(γθ1σi2l, γσi

2lθ2),

whenever the differentials of those curves exist. Furthermore, since the tangent cones over

all x ∈ Ω are Euclidean we may write

π − înt(γθ1σi2l, γσi

2lθ2) = π − ^g

σi2l

(γθ1σi2l, γσi

2lθ2)

= ^gσi2l

(γσi2lθ2, γθ1σi

2l)− π

= [^gσi2l

(γσi2lθ2, e2

⊥σi2l)− π/2] + [^g

σi2l

(e2⊥σi2l, γθ1σi

2l)− π/2] (4.18)


which is always true since we calculate the oriented angles modulo 2π. Together with

the same kind of reasoning applied to ^gσi2l

(γθ1σi2l, γσi

2lθ2) we obtain the decomposition

f 1k = f 1,+

k + f 1,−k . Now for any dyad r ∈ h · m

2l |m, l ∈ N, m ≤ 2l we apply the shift

τ : Ω → Ω, τ(x) = x− r · e2 and then the identity (4.12) of corollary 4.1 to the integrals∫Ωξ(x)f 1,+

k (x, r)dx and∫

Ωξ(x)f 1,−

k (x, r)dx respectively which gives rise to

limk→∞

∫Ω

ξ(x)f 1k (x, r)dx =

∫Ω

ξ(x− re2)1

‖e2‖gx

〈Γx(e2, e2), e2⊥x 〉gxdx

and since this limit is locally uniform w.r.t. r (see remark 4.3) we can replace r by any

s ∈ [0, h]. Invoking (4.4) once more we see that there is a constant C = Cξ such that

supk |∫

Ωξ(x)f i,±

k (x, s)dx| < C for all s ∈ [0, h] and hence we may conclude

limk→∞

h∫0

∫Ω

ξ(x)f 1k (x, s)dxds =

h∫0

∫Ω

ξ(x− se2)1

‖e2‖gx

〈Γx(e2, e2), e2⊥x 〉gxdxds

by means of Lebesgue’s dominated convergence theorem applied to the functions s →∫Ωξ(x)f 1,±

k (x, s)dx on [0, h]. Finally, e2⊥x = −(g22(x)e1 − g12(x)e2)/

√g22(x) det g(x) and

hence

1

‖e2‖gx

〈Γx(e2, e2), e2⊥x 〉gx = −Γ1

22(x)〈e1, g22(x)e1 − g12(x)e2〉gx

g22(x)√

det g(x)= −

√det g(x)

g22(x)Γ1

22(x).

In the other cases f ik(., .), i = 2, 3, 4 we argue in the same manner, noting that instead

of e⊥2 in (4.18) we take e⊥1 = −(g12e1 − g11e2)/√g11 det g for i = 2 and −e⊥i−1 for i = 3, 4.

Hence we have shown that the for k tending to Ik(h) converges to the right hand side of

(4.1), where h > 0 was chosen arbitrarily.

It remains to identify the limit of the left hand side in (4.16), where we assume first that

ξ ≥ 0. From the local equivalence of the Riemannian and Euclidean metric it is clear

that for sufficiently large k there is an ε > 0 with ε → 0 for k → ∞ such that for all

x ∈ supp(ξ) with hε = h− 2ε and xε = x+√

2ε(e1 + e2)

hε(xε) ⊂ h(x)[k, (θji )(x)]

for all possible choices of (θij) ∈ Bs3(σi

2l+1). Since we may assume w.l.o.g. that eX is

nonnegative (otherwise we argue with the measure e+X(dx) = eX(dx) + |κ|mX(dx) (c.f.

[Mac98] or section 6.2)) we may infer from this that

eX(hε(xε)) ≤ eX(h(x)[k, (θji )(x)])

from which we deduce by passing to the limit for k →∞ in (4.16) that for all ε > 0∫Ω

ξ(x)eX(hε(xε))dx ≤ limk→∞

Ik(h) =: I(h).


Hence, using the fact that hε(xε) h(x) for each x and ε→ 0 as well as the monotone

convergence theorem we obtain∫Ω

ξ(x)eX∗(h(x))dx ≤ I(h),

where eX∗ denotes the inner measure associated to eX . (An analogous discussion for

the enlarged squares hε(xε) with hε = h + 2ε and xε = x −√

2ε(e1 + e2) would yield∫Ω

ξ(x)eX(h(x))dx ≥ I(h), where eX = e∗X by definition of eX .) But now we are done

since ∫Ω

eX(h(x))ξ(x)dx ≤∫Ω

eX∗(h+ε(x))ξ(x)dx ≤ I(h+ ε)

for all ε > 0 and the continuity of (gij) implies that the function h→ I(h) is continuous.

Hence ∫Ω

eX(h(x))ξ(x)dx ≤ I(h) = limε→0

I(h− ε) ≤∫Ω

eX∗(h(x))ξ(x)dx,

which implies

eX∗(h(x)) = eX(h(x)) for dx-a.e.x ∈ Ω \Bh(∂Ω)

since eX∗ ≤ eX , i.e. h(x) is eX-measurable for dx-a.e. x ∈ Ω \Bh(∂Ω) as well as∫Ω

ξ(x)eX(h(x))dx = lim I(h) ∀ 0 ≤ ξ ∈ C0c (Ω).

which is (4.1) for non-negative ξ. For general ξ ∈ C0c (Ω) the claim follows from the

decomposition ξ = ξ+ − ξ−.

4.2 Distributional Gaussian Curvature Bounds

Proposition 4.2 (Weak lower curvature bounds). Let (X, d) be a two-dimensional

Alexandrov space with Curv(X) ≥ κ ∈ R and discrete metrically singular set SX such

that the system of natural coordinates with the associated metric tensors (gij) yield a C1,1

Riemannian structure for X \ SX , i.e. the chart transition maps are C1,1loc and the metric

tensors (gij) are locally uniformly Lipschitz continuous. Then the Gaussian curvature k

is bounded from below on U by κ in the distributional sense, i.e. let Ω = φai(U) ⊂ R2 be

the image of U ⊂ X \ SX under φai and let gai = (gij) : Ω → R2×2, gij ∈ Liploc(Ω), be

the Riemannian tensor on U with respect to φ, then for all 0 ≤ ξ ∈ C∞c (Ω) the inequality∫

Ω

[ξ,12 arccos(

g12√g11g22

)− ξ,1

√det g

g22

Γ122 − ξ,2

√det g

g11

Γ211(x)

]dx ≥ κ

∫Ω

ξ√

det g dx

holds true, where the functions Γkij(x) = 1

2gkl(x)Dxi

gjl(x) + Dxjgil(x) − Dxl

gij(x) are

obtained from the weak differentials Dxlgij of gij.


Proof. Remember that limh→0mX(h(x))/h2 =

√det gij(x) locally uniformly on Ω. Thus

if we multiply ξ(x) in the integral on the left hand side of our Gauss-Bonnet formula (4.1)

by 1 = mX(h(x))/mX(h(x)), then the result in [Mac98] that

limh→0

e(h(x))

m(h(x))≥ κ m-a.e.

together with Fatou’s lemma implies

lim infh→0

1

h2

∫Ω

ξ(x)e(h(x))dx ≥ κ

∫Ω

ξ(x)√

det gij(x)dx.

Thus it remains to compute the limit of the right hand side of (4.1) divided by h2 for h

tending to zero. From the regularity of ξ it follows easily that

limh→0

1

h2

∫Ω

h∫0

[ξ(x− te2)− ξ(x− (h− t)e2 − he1)

]√det g

g22

(x)Γ122(x)dxdt

=

∫Ω

ξ,1(x)

√det g

g22

(x)Γ122(x)dx

where ξ,1 denotes the differential of ξ with respect to the i-th direction in R2 and analo-

gously

limh→0

1

h2

∫Ω

h∫0

[ξ(x− (h− t)e1)− ξ(x− he2 − te1)

]√det g

g11

(x)Γ211(x)dxdt

=

∫Ω

ξ,2(x)

√det g

g11

(x)Γ211(x)dx.

As for the last remaining integral we may use the fact that Kx ' R2 for each x ∈ Ω once

more in order to write

4∑i=1

înti (h(x))− 2π =^x(e1, e2) + ^x+he2(−e2, e1)

+ ^x+h(e1+e2)(−e1,−e2) + ^x+he1(e2,−e1)− 2π

=^x(e1, e2) + π − ^x+he2(e1, e2) + ^x+h(e1+e2)(e1, e2)

+ π − ^x+he1(e1, e2)− 2π

=^gx(e1, e2)− ^gx+he2(e1, e2) + ^gx+h(e1+e2)

(e1, e2)− ^gx+he1(e1, e2)

which equals, since trivially sign(e1, e2) = 1, by (4.11)

= arccos(g12√g11g22

)(x)− arccos(g12√g11g22

)(x+ he2)

+ arccos(g12√g11g22

)(x+ h(e1 + e2))− arccos(g12√g11g22

)(x+ he1).


From this it is now easy to see that in fact

limh→0

1

h2

∫Ω

ξ(x)( 4∑

i=1

înti (h(x))− 2π

)dx =

∫Ω

ξ,12 arccos(g12√g11g22

)dx

which completes the proof.

Remark 4.4. For general Alexandrov spaces the assumption that gij ∈ Liploc seems

to be too strong. In fact, the weaker assumption gij ∈ BVloc(Ω) ∩ C0(Ω) would be more

natural, which follows from the results in [OS94] and [Per98], but for the reasons indicated

in remark 4.1 we have to confine ourselves to the situation stated above.

In the following corollary the Laplacian and harmonic functions on Alexandrov spaces

are understood in terms of the weak Riemannian structure, see section 5.2 or the papers

[KMS01, KS98]. On account of the distributional Gaussian curvature bounds we now

may engage the paper [CH98] which readily yields

Corollary 4.2 (Gradient Estimate). Let (X, d) an Alexandrov space with Curv (X) ≥−k2 as in proposition 4.2 then any nonnegative harmonic function u on X satisfies Yau’s

gradient estimate ∥∥∥∥∇uu∥∥∥∥

L∞(X,m)

≤√k. (4.19)

Remark 4.5. Using cut-off functions as in lemma 3.3 in [KMS01] it is obviously possible

to extend the previous statement a littler further to Alexandrov surfaces with a suffi-

ciently small exceptional set outside which the assumptions of propositions 4.2 apply. For

instance, this is the case if the singular set is discrete.

Examples 4.1.

i) Riemannian surfaces with lower sectional curvature bounds,

ii) surfaces of revolution in R2 induced by Lipschitz continuous functions f : [a, b] → R+

satisfying f ′′ ≤ k2f in a weak sense, compare [CH98] and

iii) as a special case of ii) - or iv) - one may take the double Sh ∪∂ShSh of a spherical

cap Sh = (x, y, z) ∈ S2 ⊂ R3 | z ≥ h with its induced inner metric.

iv) two-dimensional Riemannian simplicial complexes with a discrete set of 0-simplexes,

v) Alexandrov surfaces obtained by more general surgery operations, i.e. by cutting

and gluing two-dimensional Riemannian pieces of constant curvature in such a way

that metrically singular points occur only in a discrete fashion, for instance spaces

of the kind Xf or Xk of section 5.1 ex. iii) and iv) respectively,

The Heat Kernel on Alexandrov Spaces 49

5 The Heat Kernel on Alexandrov Spaces

5.1 Volume Regularity

On a smooth Riemannian manifold (Md, g) one obtains from the expansion of the volume

density function det d expx for the pull back of the Riemannian volume form on the tangent

space at xd exp∗x volM

dLn(z) = det d expx(z) = 1− 1

6Ricc(z, z) + o(|z|2). (5.1)

This formula implies in particular that both tangential and intrinsic mean value operators

can be used for the approximation of the Laplace-Beltrami operator on M:

Lemma 5.1. Let (M, g) be a smooth (i.e. C3) Riemannian manifold and f : M 7→ Rsome C3-function. Then for all x ∈M

limr→0

1

r2

[f(x)−

∫–

Br(x)

f(z)volM(dz)

]= lim

r→0

1

r2

[f(x)−

∫–

Br(0x)⊂TxM

f(expx z)dz

]= − 1

2(n+ 2)∆Mf(x).

Proof.

∫–

Br(x)

f(z)vol(dz) =1

vol(Br(x))

∫Br(0x)

f(expx(z)) det dz expx dz

=1

vol(Br(x))

∫Br(0x)

(f(x) + dfx(z) +

1

2Hessxf(z, z)− 1

6f(x) Ricc(z, z) + o(|z|2)

)dz

and hence, since∫

Br(0)A(z, z) = (n+ 2)−1r2|Br|tr(A)

=|Br|

vol(Br(x))

((1− s(x)r2

6(n+ 2))f(x) +

r2

2(n+ 2)∆f(x) + o(r2)

).

Thus

1

r2

[f(x)−

∫–

Br(x)

f(z)volM(dz)

]=

−|Br|vol(Br(x))

1

2(n+ 2)∆f(x) + θ(r)

+|Br|

vol(Br(x))

1

r2

(vol(Br(x))

|Br|−(

1− s(x)r2

6(n+ 2)

))f(x).

By virtue of the expansion of (r 7→ vol(Br(x))|Br| ) about zero (which follows from integrating

(5.1) over the ball Br(0x)) the second term on the right hand side converges to zero and

the first equality follows. The second is now also obvious.


Remark 5.1.

The same assertion is true for the analogous spherical mean value operators. Moreover,

by the Kato-Trotter formula from the preceding assertion one deduces for the family of

rescaled mean value operators

Mtf(x) :=∫–

B√t/(n+2)

(x)

f(y)m(dy) limk→∞

Mkt/kf(x) = et 1

2∆f(x)

on a smooth n-dimensional Riemannian manifold M and for sufficiently smooth f , com-

pare [Blu84]. - Similar statements can be made for the operators

Ktf(x) := (2πt)−n/2

∫M

exp(−d(x, y)2/2t)f(y)vol(dy)

Qtf(x) := (2πt)−n/2

∫M

exp(−d(x, y)2

2t+ t

s(x) + s(y)

12)f(y)vol(dy)

for which one obtains ([Sun84], [AD99] resp.)

Kkt/kf(x) → exp(t

1

2∆− 1

6s)f(x)

Qkt/kf(x) → exp(t

1

2∆)f(x)

if k →∞ for all x ∈M , where s(x) is the scalar curvature in x.

The proof of lemma 5.1 relies on the integrated version of (5.1) only, i.e. on the asymptotic

behaviour of the volume density functions qr : X 7→ R

qr(x) =m(Br(x))

bn,k(r)

for r tending to zero. Here bn,k(r) denotes the volume of an r-ball in the model space

Mn,k. This is the motivation for the following

Definition 5.1 (Volume Regularity). For α > 0 a point x in an n-dimensional Alexan-

drov space with curvature bounded below by k is said to be α-volume regular iff

1− qr(x) = o(rα) for r → 0.

(X, d) is called locally volume (Lp, α)-regular with exceptional set Sp,αX , iff Sp,α

X ⊂ X is

closed and

r−α(1− qr(.)) → 0 in Lploc(X \ Sp,α

X ) for r → 0.

Remarks 5.2.

a) The existence of the limit of m(Br(p))/bn,k(r) for r tending to zero follows from the

Bishop-Gromov volume comparison on Alexandrov spaces (cf. [Yam96]).


b) In [She93] one finds the statement that if limr→0

m(Br(x))bn,k(r)

> 12

then p is (topologically) a

manifold point. This can be seen as well from the results in [BGP92] and (5.2).

c) A volume α-regular point is also metrically regular. This is seen as follows: since

the tangent cone Kp is the pointed Gromov-Hausdorff-limit of the rescaled metric r-balls

centered at p, i.e. for all R > 0

(BrR, d

r) = (BR(p),1

rdX)

GH−→ (BR(0p), dp) for r → 0,

by theorem 10.8 in [BGP92] one obtains that the associated Hausdorff measures converge

weakly, too. Thus

mKp(B1(0p)) = limr→0

mr(Br1) = lim

r→0

m(Br(p))

rn. (5.2)

The condition of volume α-regularity implies that the limit in (5.2) equals one, and hence

Σp = ∂B1(0p) has the measure ωn and therefore must be isometric to Sn−1. - In general

the converse assertion is true only for α = 0, see example iii) below.

Examples 5.1.

i) A smooth Riemannian manifold is trivially volume (L∞, α)-regular with empty ex-

ceptional set S∞,αX for α < 2.

ii) Locally finite simplicial Riemannian complexes obtained from Riemannian simplices

of constant curvature are locally (L∞,∞)-volume-regular with S∞,∞X = SX .

iii) Take a planar circular cone Cf := B1(0)\Σf in the two-dimensional Euclidean plane,

i.e. the unit disc minus some sector Σf := (x1, x2) ∈ R2|x1 > 0, |x2| < f(x1). If f

is chosen to be convex, due to the gluing theorem [Pet97], the metric spaceXf , which

is obtained by gluing Cf (endowed with the induced Euclidean distance) along the

graph of ±f , is an Alexandrov space of nonnegative curvature. For linear f , Xf is

a flat cone which is a special case of ii). Assume f is not linear and differentiable.

Note that if f ′(0) = 0 then Xf is metrically regular. Let p = (s,±f(s)) ∈ graph(f)

⊂ Xf then for sufficiently small r the ball Br(p) ⊂ Xf is given by union of the

intersection of the Euclidean r-Balls B+r (p) (and B−

r (p)) centered at (s, f(s)) (and

(s,−f(s))) with Cf respectively and it is sufficient to compute |B+r (p) ∩ Σf |/πr2,

where |.| denotes the two-dimensional Lebesgue measure. If we shift p to the origin

and rotate the picture B+r (p)∩Σf is the intersection of the upper half r-ball around 0

with the epigraph of a convex function f such that f(0) = f ′(0) = 0. If x± = x±(r)

denote the x-coordinates of the intersection points of ∂Br(0) with the graph of

f in the first and second quadrant respectively, i.e. r = x±

√1 + (f(x±)/x±)2,


x+ > 0, x− < 0

then the estimates

1

2−∫ x+

x−f(s)ds

πr2≥ |B+

r (p) ∩ Σf |πr2

≥1

2−∫ x+

x−f(s)ds+ f(x−)(r − x−) + f(x+)(r − x+)

πr2

hold trivially and since

(r − x±)f(x±)

r2+α≈ f(x±)

|x±|1+αand

∫ x±0

f(s)ds

|x±|2+α≈ f(x±)

|x±|1+α

for small r ≈ |x±| one obtains for sufficiently small r

C1f(x+)

x1+α+

+f(x−)

|x−|1+α≤ 1

rα(1− qr(p)) ≤ C2

f(x+)

x1+α+

+f(x−)

|x−|1+α. (5.3)

For instance, if the function f has growth x1+γ around 0 then p is an α-volume

regular point iff γ > α. The function f is given as the image of f under a affine

transformation of f depending on the point p. Suppose that the estimate c1r1+γ ≤

f(r) ≤ c2r1+γ for r ∈ (−ε, ε) holds true locally uniformly with respect to p. Since the

upper estimate in (5.3) obviously applies to points in a r-neighborhood of graph(f),

one obtains that Xf is locally (Lp, α)-volume regular with empty exceptional set

Sp,αX , if α < (1 + γp)/p, in particular for S1,1

X = ∅ for any γ > 0.

iv) We sketch without proofs an idea how to construct a two-dimensional Alexandrov

space X with S1,1X 6= ∅ based on the previous example. Suppose that the lower

estimate in (5.3) holds (with a different constant C1) in the r/2-neighborhood of

graph(±f). Then the idea is to iterate the cut-off-and-glue procedure of iii) with

cusps similar to Σf such that the volume of the r-neighbourhoods of the resulting

non-volume regular points in the limit space decays like rβ with β < 1 for r tending

to zero.

To start with take the Euclidean rectangle [0, 1]× [0, L] ⊂ R2 with L > 0 and a self

similar cantor set K = [0, 1]\⋃

k∈NDk → [0, 1]×0 ⊂ [0, 1]× [0, L], where each Dk


consists of a finite number of mutually disjoint open intervals, Dk+1 ⊂ [0, 1]\⋃

i≤k Di

and⋃

k Dk is dense in [0, 1]. Then for each interval (a, b) of Dk, k ∈ N we fit a copy

Λ(a,b) of Σf into (a, b) by rotating Σf about −π/2 to point in positive y−direction

and shifting it along the (x, y)-plane into the ’gap’ (a, b), i.e. such that ∂Λ(a,b)∩y =

0 = (a, 0), (b, 0). Choosing L sufficiently large we find Λ(a,b) ∩ y = L = ∅. Let

CK,f = ([0, 1]× [0, L]) \⋃

(a,b)∈Dk, k∈N

Λ(a,b)

be the resulting Euclidean domain after removing all cusps Λ(a,b))(a,b)∈Dk, k∈N from

[0, 1]× [0, L] and denote the fractal part of its boundary by

FK,f =⋃

(a,b)∈Dk, k∈N

∂Λ(a,b) ∩ ([0, 1]× [0, L]) .

Then it follows from the construction of CK,f that for z ∈ K → [0, 1]× 0 ⊂ CK,f

and any neighbourhood Uz ⊂ R2 of z we have a lower bound for the two-dimensional

measure of the r-neighbourhood of FK,f in Uz given by

|Br (FK,f ) ∩ CK,f ∩ Uz| ≥ C3rβ ∀ r ≤ r0 (5.4)

with some β < 1 that depends on the choice of f and K.

Using Petrunin’s gluing theorem we may show that CK,f gives rise to an Alexandrov

surface XK,f with nonnegative curvature and empty boundary if we glue CK,f along

the adjacent pairs of branches of ∂Λ(a,b) ∩ ([0, 1]× [0, L]) for (a, b) ∈ Dk, k ∈ N as

well as along the two vertical boundary segments 0 × [0, L] and 1 × [0, L] and

finally attach a disk to the sphere [0, 1] × L. - Alternatively the space XK,f can

be characterized as the Gromov-Hausdorff-limit of X lK,f for l → ∞, where X l

K,f is

obtained similarly to XK,f from

C lK,f = ([0, 1]× [0, L]) \

⋃(a,b)∈Dk, k≤l

Λ(a,b)

by gluing together 0× [0, L] and 1× [0, L] and the adjacent branches of ∂Λ(a,b)∩([0, 1]× [0, L]) for (a, b) ∈ Dk, k ≤ l, by attaching a flat disk to sphere [0, 1] × Land an appropriate convex flat polygonal domain to the lower part of the boundary

∂C lK,f ∩ y = 0. Since

⋃k Dk is dense in [0, 1] the set ∂C l

K,f ∩ y = 0 → X lK,f

converges to a single point in XK,f which we denote by z. Then by (5.4) for any

neighborhood Uz ⊂ XK,f one finds that mXK,f(Br(Λ)∩Uz) ≥ C ·rβ where Λ denotes

the image of⋃

(a,b)∈Dk, k∈N ∂Λ(a,b) → XK,f and which yields together with the left

hand side of (5.3)∫Uz

1

r(1− qr(x))mXK,f

(dx) ≥ C(Uz, f)rβ · r1+γ

r2∞ for r → 0

if β + γ < 1 and hence z ∈ S1,1XK,f

6= ∅.


5.2 Dirichlet Forms and Laplacians on Metric Spaces

Recently the development of differential calculus and potential theory on general metric

spaces have gained much attention. In the framework of Alexandrov spaces a direct and

very natural construction of Sobolev spaces, Laplacians and associated diffusion processes

is possible. We present the earlier results in [Stu96, Stu98, KS01] concerning the definition

and certain properties of a canonical intrinsic Dirichlet form. The idea is simple and

based on the approximation of the gradient of a function on smooth spaces by difference

quotients. Accordingly one defines for an open subset G in X, r > 0 and a measurable

function u : X 7→ R the family of approximating Dirichlet (or energy-) forms by

ErG(u) =

C

2

∫G

∫B∗

r (x)

(u(x)− u(y)

d(x, y)

)2

mr(dy)mr(dx) (5.5)

with B∗r (x) being the punctured geodesic ball of radius r around x and the measure

mr(dx) = dx/√m(Br(x)). The constant C plays the role of a dimension, but can be

chosen arbitrarily. The generator of this form

Aru(x) = ρr(x)

u(x)− ∫X

u(y)σr(x, dy)

with the Markov transition kernel

σr(x, dy) =1

sr(x)

1

d2(x, y)

1√m(Br(y))

1‖‖B∗r (x)(y)m(dy)

sr(x) =

∫B∗

r (x)

1

d2(x, y)

1√m(Br(y))

m(dy), ρr(x) =Csr(x)√m(Br(x))

is a non-local bounded symmetric operator on L2(G,m) giving rise to the L2-semigroup

P rt = e−tAr whose associated continuous time Markov jump process (Ξr

t )t≥0 is character-

ized by the one-step transition function σr(x, dy) and the exponential jump rate ρr(x) at

a given point x ∈ X. In [Stu98] it is shown that if some metric measure space (X, d,m)

possesses the so called measure contraction property, which is equivalent to lower Ricci

curvature bounds in the smooth Riemannian case, the forms of type (5.5) converge to

a limiting form E for r tending to zero in the Γ-sense, which in particular preserves

the Dirichlet form properties for the limiting functional EG and hence we obtain an m-

symmetric Hunt process Ξ generated by EG.

Moreover, since the convergence of ErG is in fact a little better, namely in the stronger

sense of Mosco (cf. [Mos94]). the semigroups P rt converge strongly to Pt, which means

that the processes Ξr convergence to Ξ in the sense of finite dimensional distributions.

Thus the limiting form E generates a process which is the scaling limit of a family of


intrinsic jump process on (X, d,m) whose data are just the metric d and the measure m.

Kuwae and Shioya [KS01] modified this result for forms of the type

Eb,rG (u) =

n

2

1

bn,k(r)

∫G

∫B∗

r (x)

(u(x)− u(y)

d(x, y)

)2

m(dy)m(dx)

defined on Alexandrov spaces with lower curvature bound, for which they introduced the

notion of generalized measure contraction property (see [KS01] for details). Also, they

proved that on Alexandrov spaces in the limit both approximating forms Eb,rG and Er

G (for

C = n) give the same result (ibid. Corollary 5.1). Summing up these results one obtains

the following assertions:

Theorem 5.1 (Existence and uniqueness of the canonical Dirichlet form). On

an n-dimensional Alexandrov space with lower curvature bound k and some G ⊂ X with

m(G) < ∞ both sequences ErG and Eb,r

G have Γ-limits on L2(G) which coincide with their

common pointwise limit EG on Lip(X). The L2(G)-closure (EG,D(EG)) of EG on Lip(X)

is a strongly local and regular Dirichlet form on L2(G).

The generator ∆G of this form will be called Laplacian for it coincides (up to a trivial

time scaling) with the Neumann Laplace-Beltrami operator if X is smooth (see lemma

5.1). Moreover, since the measure m is doubling and a local weak Poincare inequality

applies, by Moser iteration one can show the existence and Holder continuity of the heat

kernel qGt for the corresponding semigroup [Stu96, Stu98]. In particular the semigroup

has the Feller property, which will be used later on. Also, the analogous results hold true

if one considers the Dirichlet Laplacian instead which is defined as the generator of the

closure of (E ,Lipc(G)). For the sake of completeness we should mention also the approach

in [KMS01] for the definition of the Laplacian on Alexandrov spaces, where the energy

form E is defined just as in the smooth case by employing the weak Riemannian structure

of (X, d). - In any case the m-symmetric Hunt process on X which is associated with

E is called canonical because of its close and unique relation with the metric d (see also

section 5.7).

For the comparison with the Riemannian case we introduce a third sequence of oper-

ators and corresponding forms which will be useful later and which are given by

AE,ru(x) = nm(Br(x))bn,k(r)

1r2

(u(x)−

∫–

Br(x)

u(y)m(dy)

)EE,r

G (u) = n2bn,k(r)

∫G

∫Br(x)

|u(x)−u(y)|2r2 m(dx)m(dy).


Lemma 5.2. On Lip(G) the limit of EE,rG for r → 0 exists and coincides with n

n+2EG.

Proof. We compare EE,rG with Eb,r

G . For this purpose define for α ∈ (0, 1)

EE,rG,Bα

(u) = n2bn,k(r)

∫G

∫Bαr(x)

|u(x)−u(y)|2r2 m(dx)m(dy)

EE,rG,Aα

(u) = n2bn,k(r)

∫G

∫Br(x)\Bαr(x)

|u(x)−u(y)|2r2 m(dx)m(dy)

and Eb,rG,Bα

and Eb,rG,Aα

analogously. Then we obtain

Eb,rG (u) = Eb,r

G,Bα(u) + Eb,r

G,Aα(u)

≤ Eb,rG,Bα

(u) +1

α2EE,r

G,Aα(u)

= Eb,rG,Bα

(u)− 1

α2EE,r

G,Bα(u) +

1

α2EE,r

G (u)

=bn,k(rα)

bn,k(r)Eb,αr

G (u)− bn,k(rα)

bn,k(r)EE,αr

G (u) +1

α2EE,r

G (u).

Due to the convergence of Eb,rG (u) and limr→0

bn,k(rα)

bn,k(r)= αn in the limit this yields

EG(u) ≤ 1

α2(1− αn)lim inf

r→0

(EE,r

G (u)− α2 bn,k(rα)

bn,k(r)EE,αr

G (u)

)≤ 1

α2(1− αn)

(lim inf

r→0EE,r

G (u)− αn+2 lim infr→0

EE,αrG (u)

)=

1− αn+2

α2(1− αn)lim inf

r→0EE,r

G (u).

As limα→11−αn+2

α2(1−αn)= n+2

nwe see EG(u) ≤ n+2

nlim inf

r→0EE,r

G (u) if we send α to 1. The reverse

inequality can be proved in a similar way. One writes

EE,rG (u) = EE,r

G,Bα(u) + EE,r

G,Aα(u)

≤ α2bn,k(rα)

bn,k(r)EE,αr

G (u) + Eb,rG (u)− bn,k(rα)

bn,k(r)Eb,αr

G (u).

Hence

(1− αn)EG(u) = limr→0

(Eb,r

G (u)− bn,k(rα)

bn,k(r)Eb,αr

G (u)

)≥ lim sup

r→0

(EE,r

G (u)− α2bn,k(rα)

bn,k(r)EE,αr

G (u)

)≥ (1− αn+2) lim sup

r→0EE,r

G (u).

Consequently, upon dividing this inequality by (1−αn) and letting tend α to 1 one obtains

EG(u) ≥ n+2n

lim supr→0

EE,rG (u) and the claim follows.


5.3 Laplacian Comparison

The main result of this section is the comparison theorem for ∆X which is a generalization

of the well known Laplacian comparison theorem for Riemannian manifolds with lower

Ricci curvature bound. Since a precise characterization of Dom(∆X) is not available, the

inequality for ∆X is stated in the weak form, which involves the Dirichlet form E which

is generated by ∆X :

Theorem 5.2 (Laplacian Comparison). Let X be an n-dimensional Alexandrov space

with curvature bounded below by k ∈ R which is locally volume (L1, 1)-regular with ex-

ceptional set S1,1X of rough dimension ≤ n − 2. Then for any f ∈ C3(R) with f ′ ≤ 0,

0 ≤ ζ ∈ D(E) and p ∈ X

E(f dp, ζ) ≤ 〈−S1−nk (Sn−1

k f ′)′ dp, ζ〉L2(X,m) (5.6)

where dp(x) := d(p, x) and

Sk(t) =

1/√k sin(

√kt) if k > 0

t if k = 0

1/√

(−k) sinh(√−kt) if k < 0.

Proof. The proof is based upon the choice of a suitable metric on each tangent cone

Kx, x ∈ X and a modification Λx : X 7→ Kx of the inverse of the exponential map

which allows to apply the Alexandrov convexity of (X, d) - For each x ∈ X we equip the

tangent cone Kx over x with the hyperbolic, spherical or flat metric dxk defined by the

corresponding cosine law, i.e.

cosh(√−kdx

k[(α, s), (β, t)]) = cosh(√−ks) cosh(

√−kt)

− sinh(√−ks) sinh(

√−kt) cos d^(α, β)

cos(√kdx

k[(α, s), (β, t)]) = cos(√ks) cos(

√kt)

− sin(√ks) sin(

√kt) cos d^(α, β)

(dxk[(α, s), (β, t)])

2 = s2 + t2 − 2st cos d^(α, β)

depending on whether k < 0, k > 0 or k = 0 respectively in order to obtain a new

curved tangent cone Kx,k =((Σx × R+)/ ∼dx

k, dx

k

)∼, which will be denoted (Mx, d

x). Let

Mn,k(X) =⋃

x∈XMx denote the corresponding curved tangent cone bundle and note that

(Mx, dx) w Mn,k for each x ∈ X \ SX . Then the map Λx : X 7→ Mx is chosen as the

canonical radially isometric extension of (some choice of) the map

λx : X 7→ Σx λx(z) = γ′xz(0)

which is the projection of z onto one of the directions by which it is seen from x. It is

possible to choose λx in such a way that the map Λ : X×X 7→ Mn,k(X), (x, z) 7→ Λx(z) ∈


Mn,k(x) is measurable, where Mn,k(X) is endowed with the product sigma algebra. (Also

there is a natural measure on Mn,k(X) as a product of m and the n-dimensional Hausdorff

measure in each fiber.) Now Curv(X) ≥ k implies that for x ∈ X \ SX the map Λx is

expanding. In fact, for y, z ∈M let y, z be the image points of y, z under Λx and denote

0x = Λx(x), then by construction of Λx

dx(0x, y) = d(x, y), dx(0x, z) = d(x, z)

^(γxy, γxz) = d^(γ′xy, γ′xz) = ^(γ0xy, γ0xz)

where γ0xy and γ0xz denote the uniquely defined geodesics in Mx joining 0x with y and z

respectively. Since Mx w Mn,k one obtains

dx(Λx(y),Λx(z)) ≥ d(y, z) (5.7)

because the contrary would mean a contradiction to the Alexandrov convexity for geodesic

hinges (proposition 3.1) in the global version.

Let now be f a function as required and p ∈ X. Then f dp is Lipschitz and thus in D(E).

Let us first assume that the nonnegative test function ζ ∈ D(E) has compact support.

On account of the lemma 5.2 and the polarization identity we know that

〈Ar(f dp), ζ〉L2(X,m) = EE,r(f dp, ζ) →n

n+ 2E(f dp, ζ) for r → 0.

If px denotes the image point of p under Λx the monotonicity property of f together with

(5.7) yields

Ar(f dp)(x) ≤ qr(x)n

r2

(f(dx(px, 0x))−

∫–

Br(x)

f(dx(px,Λx(y)))m(dy)

)

with qr(x) = m(Br(x))bn,k(r)

. Also by (5.7) the image measure of m under Λx on Mn,k(x) is

absolutely continuous with respect to the volume measure volx on Mx

d ((Λx)∗m)

dvolx=: ρx ≤ 1 volx-a.e.

Thus by the general integral transformation formula

Ar(f dp)(x) ≤qr(x)n

r2

(f(dx(px, 0x))−

1

qr(x)

∫–

Br(0x)

f dxpx

(z)ρx(z)volx(dz)

)

= qr(x)n

r2

(f(dx(px, 0x))−

∫–

Br(0x)

f dxpx

(z)volx(dz)

)

+ qr(x)n

r2

∫–

Br(0x)

f dxpx

(z)

[1− ρx(z)

qr(x)

]volx(dz). (5.8)


As S1,1X has rough dimension ≤ n−2 from lemma 3.3 in [KMS01] it follows that we can find

a sequence of cut-off functions in D(E) vanishing on some neighborhood of(p ∪ S1,1

X

)∩

supp(ζ) and converging to the constant function 1 in the Dirichlet space (D(E), ||.||E1 ). So

we may assume that ζ is zero on some neighborhood of p ∪ S1,1X .

The volume regularity of X implies that qr(x) → 1 for (some subsequence if necessary)

r → 0 m-a.e. on supp(ξ) and thus by lemma 5.1 and the special structure of the Laplace-

Beltrami operator ∆Mn,k acting on radial functions one obtains for the first term on the

right hand side of (5.8) and x ∈ supp(ξ)

limr→0

qr(x)n

r2

(f(dx(px, 0x))−

∫–

Br(0x)

f dxpx

(z)ρx(z)volx(dz)

)= − n

n+ 2∆Kx(f dx

px)(0x) =

n

n+ 2S1−n

k (Sn−1k f ′)′ dpx(0x)

=n

n+ 2S1−n

k (Sn−1k f ′)′(d(p, x)).

For the second term on the right hand side of (5.8) note first that we may assume without

loss of generality that f(0) = 0. Then by the regularity of f

n

r2

∫–

Br(0x)

f dxpx

(z)

[1− ρ(z)

qr(x)

]vol(dz)

=n

r2

f ′(dp(x))

dp(x)

∫–

Br(0x)

〈px, z〉[1− ρ(z)

qr(x)

]vol(dz)

+1

2

n

r2

∫–

Br(0x)

Hess0x [f dxpx

](z, z) + of (|z|2)[1− ρ(z)

qr(x)

]vol(dz).

Using the relation 1− ρ(z)qr(x)

= (1− q−1r (x))(1− 1−ρ(z)

1−qr(x)) and rescaling the integrals we see

that the second term in the right hand side of (5.8) equals

nqr(x)− 1

r

f ′(dp(x))

dp(x)

∫–

B1(0x)

〈px, z〉1− ρ(rz)

1− qr(x)vol(dz) +

n

2(1− qr(x))

∫–

B1(0x)

Hess0x [f dxpx

](z, z)

[1− 1− ρ(zr)

1− qr(x)

]vol(dz) + ϑf (r)

≤ nf ′(dp(x))qr(x)− 1

r+ n(1− qr(x))C(f ′′, supp(ζ)) + ϑf (r)

with ϑf (r) → 0 for r → 0. Thus we get the desired inequality for compactly supported

ζ from the local (L1, 1)-volume regularity if we multiply (5.8) by ζ, integrate over X and

let r tend to zero.

For general ζ ∈ C0(X) ∩ D(E) take a sequence of smooth nonnegative functions with

compact support ηk : R+ 7→ [0, 1] such that η(t) = 1 for t ∈ [0, k] and set ζk = ζ · ηk dp


in order to obtain (5.6) for ζk. The Dirichlet form E is strongly local and hence the

corresponding energy measure µ〈.,.〉, which is a measure valued symmetric bilinear form

on D(E) defined by∫X

φ(x)µ〈u,u〉(dx) = 2E(u, φu)− E(u2, φ) ∀φ ∈ C0(X)

has the derivation property

dµ〈u·v,w〉 = udµ〈v,w〉 + vdµ〈u,w〉

for all u, v and w ∈ D(E) (cf. [FOT94], section 3.3.2). Also from the construction of Eit is obvious that µ〈u,v〉 m for u, v ∈ D(E) and thus E admits the Carre du Champ

operator Γ which is defined via the corresponding density, i.e.

Γ(u, v) :=dµ〈u,v〉

dm∈ L1(X,m) ∀u, v ∈ D(E)

and which yields the representation E(u, v) =∫

XΓ(u, v)dm for u, v ∈ D(E). Hence a

twofold application of Lebesgue’s theorem and limk→∞ Γ(f dp, ηk dp) = 0 m-a.e. yield

E(f dp, ζ) =

∫X

Γ(f dp, ζ)dm

= limk→∞

∫X

ηk dpΓ(f dp, ζ)dm+ limk→∞

∫X

ζΓ(f dp, ηk dp)dm

= limk→∞

∫X

Γ(f dp, ηk dp · ζ)dm = limk→∞

E(f dp, ζk)

≤ − limk→∞

〈S1−nk (Sn−1

k f ′)′ dp, ζk〉L2(X,m)

= −〈S1−nk (Sn−1

k f ′)′ dp, ζ〉m.

The weak inequality (5.6) becomes a pointwise bound if f dp ∈ D(∆X), in which case

the classical result is completely recovered:

Corollary 5.1. If f dp ∈ D(∆X) for some p ∈ X and non-increasing f ∈ C3(R) then

∆X(f dp)(x) ≥ S1−nk (Sn−1

k f ′)′ dp(x) for m-a. a. x in X.

Remark 5.3. Probably one might think about a different approach to proving a com-

parison theorem for the Laplacian on Alexandrov spaces via some sort of second variation

formula for arclength or generalized Jacobi fields, compare [Ots98] for some work in this

direction. However, such an approach seems to require much more work on geodesics on

Alexandrov spaces and is probably less extendable to more general situations than the

proof given above.


Remark 5.4. Under the weaker assumption that 1− qr(x) ≤ O(r) locally uniformly on

X \ SX one obtains an additional drift term in the Laplacian comparison principle which

then takes the form

E(f dp, ζ) ≤ (−S1−nk (Sn−1

k f ′)′ dp, ζ)L2(X,m)

− (n+ 2) supν

∫Mn,k(X)

f ′(dp(x))

dp(x)〈px, z〉ζ(x)ν(dx, dz) (5.9)

where supremum with respect to ν is taken over all weak accumulation points of the

weakly precompact sequence of measures on Mn,k(X)

νr(dz, dx) =1− qr(x)

r

1− ρx(rz)

1− qr(x)1‖‖B1(0x)

(z)volx(dz)m(dx).

If z 7→ ρx(z) is differentiable in 0x we find

νr(dx, dz) → ν(dx, dz) = dρx(z)1‖‖B1(0x)(z)volx(dz)m(dx)

and the drift part in (5.9) becomes∫X

f ′(dp(x))

dp(x)dρx(px)ζ(x)m(dx).

However, the drift term in (5.9) can be interpreted as a measure for the local approxima-

tion of X by its tangent spaces.

5.4 Heat Kernel Comparison

The following paragraphs contain a number of corollaries to the weak Laplacian compar-

ison, the first concerning the associated Dirichlet heat kernel. For this recall that the

Dirichlet Laplacian is the generator of (EΩ, Dc(EΩ)) obtained from taking the closure of

the set of compactly in Ω supported Lipschitz functions with respect to the Dirichlet norm

‖.‖21 = ‖.‖2

L2(Ω) + EX(.). As usual the fundamental solution qGt of the corresponding heat

equation is called Dirchlet heat kernel.

Theorem 5.3. Let qGt be the Dirichlet heat kernel on some domain G ⊂ X and let

x 3 Br(x) ⊂ X. Then for all y ∈ Br(x) and x, y ∈ Mn,k with d(x, y) = d(x, y)

qGt (x, y) ≥ qk,r

t (x, y) (5.10)

where d denotes the distance on Mn,k and qk,rt is the Dirichlet heat kernel of Br(x) ⊂ Mn,k.

Example 5.2. Consider the heat kernel qCf

t on Cf (where Cf is defined as in example

iii) of section 5.1) satisfying the boundary conditions

f = 0 on S1 ∩ ∂Cf ,∂u

∂ν(x, f(x)) = −∂u

∂ν(x,−f(x)) on ∂Σf


where ∂u∂ν

is the exterior normal derivative of u. The conditions on ∂Σf are chosen in

such a way that solving this boundary value problem is consistent with gluing the two

half-sectors together along the graph of ±f . By the heat kernel comparison theorem we

now get a lower bound for the flat heat kernel qCf

t on Cf of the form qCf

t (x, y) ≥ qt(x, y),

where qt is the Euclidean Dirichlet heat kernel on B1.

The proof of theorem 5.3 is an application of theorem 5.2 to the heat kernel on the model

space together with following simple version of a parabolic maximum principle for ∆X :

Lemma 5.3. Let f : Ω × (0, T ) → R with f ∈ L2([0, T ], Dc(EΩ)) ∩ C([0, T ]), L2(Ω)). If

f0 := f(0, .) ≤ 0 m-a. e. and Lf ≥ 0 with L = ∆Ω − ∂t in the following weak sense

τ∫σ

E(f(t, .), ξ)dt ≤ − 〈f(.), ξ〉m|τσ (5.11)

for all σ, τ ∈ (0, T ) and 0 ≤ ξ ∈ Dc(EΩ), then f(t, x) ≤ 0 for m-a.e. x ∈ Ω and t ∈ [0, T ].

Proof. For ε > 0 consider fε(t, x) = 1ε

t+ε∫t

f(s, x)ds. Then fε is a subsolution to the heat

equation in the following sense: for all nonnegative ξ ∈ L2([0, T ], Dc(EΩ)) and σ, τ ∈(0, T − ε):

τ∫σ

E(fε(t, .), ξ(t))dt =

τ∫σ

1

ε

t+ε∫t

E(f(s, .), ξ(t))dsdt

≤ −τ∫

σ

1

ε〈f(t+ ε, .)− f(t, .), ξ(t, .)〉mdt

= −τ∫

σ

〈∂tfε(t, .), ξ(t, .)〉mdt

Now we take ξ to be j(fε) where j ∈ C∞(R) with j′ ≥ 0, ‖j′‖∞ < ∞ and j = 0 on

(−∞, δ], j > 0 on (δ,+∞) for some δ > 0. From the definition of Dc(EΩ) it follows that

this choice of ξ is admissible. Inserting this into the last inequality yields

0 ≤τ∫

σ

∫Ω

j′(fε(t, x))µ〈fε(t)〉(dx)dt ≤τ∫

σ

〈∂tfε(t), j(fε)〉m︸︷︷︸=∂t〈J(fε(t))〉m

dt = 〈J(fε(σ))〉m − 〈J(fε(τ))〉m

with J(t) =∫ t

0j(s)ds. Using fε(σ) → fε(0) and fε(0) → f0 in L2(Ω) as well as J(f0) = 0

m-a.s., by sending first σ → 0 and then ε → 0 we find that 〈J(f(σ))〉m ≤ 0. From the

definition of j and J resp. this implies that f(t, x) ≤ δ for m a.e. x ∈ Ω. Sending δ → 0

yields the claim.


Proof of theorem 5.3. Let us first assume that Br(x) ⊂ X. If qk,rt (. , .) denotes the Dirich-

let heat kernel on Br(x) ⊂ Mn,k then there is the uniquely defined real valued function

(t, s) 7→ ht(s) = hk,rt (s) satisfying the differential equation ∂tht(s) = −S1−n

k ∂s(Sn−1k ∂sht(s))

for (t, s) ∈ R+ × (0, r) and such that qk,rt (x, y) = ht(d(x, y)) (see [Cha84]). Furthermore,

since s 7→ ht(s) is non-increasing (ibid., lemma 2.3), the continuation of ht (still denoted

by ht(s))

R+ × R+ 3 (t, s) → hk,rt (s) =

ht(s) for s ∈ [0, r]

0 for s > r

is locally Lipschitz in both variables, non-increasing in s and satisfies

∂tht(s) ≥ −S1−nk ∂s(S

n−1k ∂sht(s)) (5.12)

in the distributional sense. In order to prove (5.12) it is sufficient to note that for the

Laplacian ∆Mn,k of the function qk,r,xt : R+ ×Mn,k → R+, (t, y) → ht(d(x, y)) one finds

−∆Mn,kqk,r,xt = −∆

Mn,ky qk,r

t (x, .)1‖‖Br(x) +∂qk,r

t (x, .)

∂ν|∂Br(x)

dHn−1|∂Br(x)

in the distributional sense, where the density in front of the Hausdorff-measure is obviously

non-positive. Testing this inequality with radially symmetric test functions and using the

special form of ∆Mn,k we obtain (5.12). Hence, if we mollify (t, s) → ht(s) with respect to

s by a non-negative smooth kernel we obtain a family (t, s) → hρt (s) of smooth functions,

non-increasing in s, satisfying (5.12) in a pointwise sense and which converge to h locally

uniformly on R+ × R+ for ρ → 0, which in particular implies that for all T > 0 and ρ

sufficiently small (depending on T ) the function

G 3 y → ψρt (y) = hρ

t (d(x, y))

belongs to Dc(E(G)) for all t ∈ (0, T ). Using the weak Laplacian comparison inequality

(5.6) and (5.12) one deduces that (t, y) → ψρt (y) satisfies (5.11) for 0 < σ ≤ τ for

sufficiently small ρ and all 0 ≤ ξ ∈ Dc(EG). If we assume also that ξ ∈ D(∆G) then we

may integrate by parts on the left hand side of (5.12), pass to the limit for ρ → 0 and

integrate by parts again which yields for the function ψt(y) = ht(d(x, y))

τ∫σ

E(ψt, ξ)dt ≤ −〈ψτ , ξ〉m + 〈ψσ, ξ〉m ∀ 0 ≤ ξ ∈ D(∆G). (5.13)

Standard arguments of Dirichlet form theory now show that for ξ ∈ D(E) the sequence

ξλ = λRλξ ∈ D(∆G) converges to ξ in the Dirichlet space (D(E), ‖.‖1) for λ tending to

infinity, where Rλ is the λ−resolvent associated to (E , D(E)). From the representation

Rλ = λ∫∞

0e−λsPsds and the fact that the heat semigroup is positivity preserving ξ ≥ 0

implies ξλ ≥ 0. Hence we obtain (5.13) for arbitrary 0 ≤ ξ ∈ Dc(EG) by approximation.


For δ > 0 let ψδt (y) = (PG

ρ ψt)(y) =∫

GqGρ (y, z)ψt(z)m(dz) and qG,δ

t (y) = qGt+δ(x, y). Then

ψδt and qG,δ

t belong to Dc(EG), where qG,δt obviously satisfies (5.11) with equality sign,

whereas for ψδt (y) we find

τ∫σ

EG(ψδt , ξ)dt =

τ∫σ

EG(ψt, PGδ ξ)dt

≤ −〈ψτ , PGδ ξ〉m + 〈ψσ, P

Gδ ξ〉m

= −〈ψδτ , ξ〉m + 〈ψδ

σ, ξ〉m ∀ 0 ≤ ξ ∈ Dc(E(G)).

Hence the function (t, y) → ft(y) = ψδ(y) − qG,δt (y) satisfies (5.11). As for the ini-

tial boundary value f0(.) we study the behaviour of ψt(.) when t tends to zero: as be-

fore the Alexandrov convexity implies that there is a radially isometric (i.e. d(x, y) =

d(Λx(x),Λx(y)) for all y ∈ X) and non-expanding map Λx : (X, d) 7→ (Mn,k, d). Thus

limt→0

∫X

ht(d(x, y))m(dy) = limt→0

∫Br(Λx(x))

ht(d(Λx(x), y))Λx∗m(dy)

=d(Λx∗m)

d volMn,k

(Λx(x)) = limr→0

m(Br(x))

bn,k(r)=: q0(x)

(with q0(x) ≤ 1 and m (q0 < 1) = 0). From this and the standard Gaussian estimates

for the heat kernel on smooth manifolds it follows that for any ζ ∈ C0c (X)

limt→0

∫X

ht(d(x, y))ζ(y)m(dy) = Cxζ(x)

with Cx = 1/q0(x), i.e. the function y 7→ ht(d(x, y)) converges weakly to Cxδx for t

tending to zero. From the local (L1, 1)-volume regularity of (X, d) it follows that Cx = 1

for m-almost every X. Hence let us assume that Cx = 1 for the previously fixed x. Then

it follows that in fact ft = ψδt − qG,δ

t converges to f0(y) = qGδ (x, y) − qG

δ (x, y) = 0 in

L2(G, dm) for t → 0 and we may apply lemma 5.3 which yields ft(.) ≤ 0 m-almost

everywhere in G. Since δ > 0 was arbitrary and PGδ → 1 in L2(G) for ρ → 0 this

implies also ψt(y) ≤ qGt (y) for m-almost every y ∈ G. Hence from the continuity of

ψt(.) = ht(d(x, .)) and qGt (.) = qt(x, .) we may conclude

qk,rt (x, y) = ht(d(x, y)) ≤ qG

t (x, y) ∀ y ∈ Br(x). (5.14)

Since the set R = x ∈ G|Cx = 1 is dense in G for general x ∈ G with Br(x) ⊂ G and

y ∈ Br(x) we may find an approximating sequence (xl, yl) with d(xl, yl) = d(x, y) = d(x, y)

such that yl ∈ Br(xl) b G and Cxl= 1, which by the continuity of qG

t (. , .) establishes

(5.14) also in the case Cx > 1. Finally, we can offset the assumption Br(x) b G by

considering Br−ε(x) first which gives (5.14)r′ for r′ = r − ε and fixed y ∈ Br−ε(x). Using

the continuity of the Dirichlet heat kernel qrt (x, y) on Br(x) ⊂ Mn,k with respect to r


(which follows from the parabolic maximum principle on Mn,k) we may pass to the limit

for ε→ 0 in the left hand side of (5.14), obtaining

qk,rt (x, y) = ht(d(x, y)) ≤ qG

t (x, y) ∀ y ∈ Br−ε(x)

where ε > 0 is arbitrary. Hence, for general y ∈ Br(x) the claim follows from the continuity

of qGt (x, .) by an approximation with Br−1/l(x) 3 yl → y for l→∞.

5.5 Eigenvalue Comparison

An immediate corollary to the heat kernel comparison theorem is the analogue of Cheng’s

eigenvalue comparison theorem [Che75]:

Theorem 5.4. Let (X, d) be a (L1, 1)-locally volume regular n-dimensional Alexandrov

space with lower curvature bound k and dimr S1,1X ≤ n− 2. Then for any ball Br(x) ⊂ X

its first Dirichlet eigenvalue λ1(Br(x)) is bounded from above by

λ1(Br(x)) ≤ λk1(r)

where λk1(r) denote the first Dirichlet eigenvalue for Br(0) ⊂ Mn,k.

Proof. This follows directly from the heat kernel comparison theorem and the eigenfunc-

tion expansion of the heat kernel on X ([KMS01]) and on Mn,k, c.f. [SY94].

Corollary 5.2. Let (X, d) be as above and for r > 0 let p ∈ X such that X0 := Br(p) ⊂ X.

If λj(X0) denotes the j-th (counted with multiplicity) Dirichlet eigenvalue of Bj(p) with

0 = λ0(X0) < λ1(X0) ≤ λ2(X0) ≤ · · · . Then

λj(X0) ≤ λk1( diam(X0)/2j).

This is a consequence of the Max-Min-principle for the higher eigenvalues of the Laplace

operator. For further standard results in this direction see [Cha84], chapter III.

5.6 Distance and Short Time Asymptotics

Another consequence of the heat kernel comparison theorem is the extension of Varadhan’s

formula [Var67], which relates the distance on a manifold to the asymptotic behaviour of

the heat kernel for short time, to Alexandrov spaces (X, d):

Proposition 5.1. Let G ⊂ X and qGt (., .) be the (Dirichlet of Neumann) heat kernel on

G under the same conditions of theorem 5.3. The for all x, y ∈ G

limt→0

2t log qGt (x, y) = −d2(x, y). (5.15)


Proof. This follows from the Davies’ sharp upper Gaussian estimate which persists on

local Dirichlet spaces with Poincare inequality and doubling base measure (cf. [Stu95]).

The other inequality follows from (5.10) and (5.15) on manifolds with lower Ricci bound

(cf. [Dav89]) applied to qk,rt on Mn,k .

Remark 5.5. In [Nor96] Norris extends Varadhan’s formula to the case of Lipschitz

(Riemannian) manifolds with measurable and uniformly elliptic metric tensor (gij), which

rules out the most general Alexandrov spaces. However, since he does not impose any

curvature condition on the resulting metric space (X, d) his result is rather complementary

and not just a special case of ours.

5.7 Diffusion Process Comparison

In the classical situation where Ξ ∈M is some semi-martingale on a smooth Riemannian

manifold (Md, g) and f ∈ C2(M ; R) the geometric Ito-formula yields for the composite

process f(Ξ) the representation

d(f Ξ) =d∑

i=1

df(Ξ)(Uei)dZi +

1

2

d∑i,j=1

(∇df)(Ξ)(Uei, Uej)d[Zi, Zj]

where U ∈ O(M) is the horizontal lift of Ξ onto the orthonormal frame bundle of (M, g)

and Z ∈ Rd is the stochastic anti-development of Ξ (see, for instance, [HT94]). In

particular if Ξ is a Brownian Motion on M this formula reduces to

d(f Ξ) = df(Uei)dWi +

1

2∆f(Ξ)dt

with W ∈ Rd being a Brownian Motion on Rd and ∆ the Laplace-Beltrami Operator

on (M, g). In this section a decomposition of the same type will be established for the

process ρp(Ξ), where Ξ is the Hunt process generated by the Dirichlet form (E , D(E))

on the Alexandrov space (X, d). We start with two general observations concerning the

martingale and zero energy part in the Fukushima decomposition of the Dirichlet process

df(Ξ):

Lemma 5.4. Let (E , D(E)) be a strongly local symmetric Dirichlet form on some Hilbert

space L2(X, σ,m) and f ∈ D(E). Then for arbitrary g, h ∈ L2 ∩ L∞(X,m) such that

gf2 ∈ D(E)

Eh·m(g(ΞS)〈M [f ]〉S

)=

S∫0

∫X

Pth · PS−tgµ〈f〉dt

where 〈M [f ]〉 is the quadratic variation process of the martingale additive functional part

of df(Ξ) and µ〈f〉 is the energy measure associated to f .


Proof. This fact essentially follows from [FOT94], theorem 5.2.3. and lemma 5.1.10.

However, we present here an almost self contained proof which requires only a certain

familiarity with the concept of energy measures. For ∆ ∈ R small let S, T be some fixed

positive numbers such that S > T + ∆. For general sufficiently regular g, h the Markov

property of Ξ yields

Eh·m(g(ΞS)[f(ΞT+∆)− f(ΞT )]2)

= Eh·m(EΞT+∆(g(ΞS−(T+∆))[f(ΞT+∆)− f(ΞT ))]2)

= Eh·m((PS−(T+∆)g)(ΞT+∆)[f(ΞT+∆)− f(ΞT ))]2)

= EPT h·m((PS−(T+∆)g)(Ξ∆)[f(Ξ∆)− f(Ξ0))]2). (5.16)

We want to divide in (5.16) by ∆ and pass to the limit for ∆ tending to zero. Before

doing so one notices that in general for f, g ∈ D(E)

lim∆→0

1

∆Eh·m(g(Ξ∆)[f(Ξ∆)− f(Ξ0)]

2) =

∫X

h · gµ〈f〉. (5.17)

If in (5.17) the term g(Ξ∆) was replaced by g(Ξ0) this would just be the well known

coincidence of the energy measure µ〈f〉 and the Revuz-measure µ〈M [f ]〉 ([FOT94], lemma

5.3.3.), but in the given form (5.17) can be verified as a consequence of the chain rule for

the energy measure, which holds true by the strong locality of E . In a second step one

has to verify that whenever hf2, g ∈ D(E)

lim∆→0

1

∆Eh·m(P∆g(Ξ∆)[f(ΞT+∆)− f(ΞT )]2)

− lim∆→0

1

∆Eh·m(P∆g(Ξ∆)[f(ΞT+∆)− f(ΞT )]2)

= lim∆→0

1

∆〈P∆h, [P∆g − g]f 2〉m

− lim∆→0

2〈P∆(fh), [P∆g − g]f〉m + lim∆→0

〈P∆(f 2h), [P∆g − g]〉m

= E(hf2, g)− 2E(hf2, g) + E(hf2, g) = 0.

Due to these two assertions taking the limit in (5.16) gives

lim∆→0

1

∆Eh·m(g(ΞS)[f(ΞT+∆)− f(ΞT )]2) =

∫X

PTh · PS−Tgµ〈f〉.


Now it is easy to compute the quadratic variation of M [f ] because we know that by general

Ito theory

Eh·m(g(ΞS)〈M [f ]〉S) = lim∆→0

Eh·m(g(ΞS)〈M [f ]〉S−∆)

= lim∆→0

Eh·m(g(ΞS)

b S∆c−2∑i=0

[f(Ξ(i+1)∆ − f(Ξi∆)]2)

= lim∆→0

b S∆c−2∑i=0

(∆

∫X

Pi∆h · PS−i∆gµ〈f〉 +O(∆))

=

S∫0

∫X

Pth · PS−tgµ〈f〉dt.

Corollary 5.3. Let (E , D(E)) be a strongly local Dirichlet form defined on L2(X, σ,m)

such that the associated semigroup has the Feller property. Then for f ∈ D(E) the fol-

lowing implication holds:µ〈f〉 = m

=⇒M 〈f〉 is a real Px-Brownian Motion for all x ∈ X.

Proof. For µ〈f〉 = m the previous lemma gives

Eh·m(g(ΞS)〈M [f ]〉S

)=

S∫0

〈Pth, PS−tg〉mdt

= S〈h, PSg〉m = SEh·m(g(ΞS)).

By a monotone class argument this implies 〈M [f ]〉S = S Ph·m-almost surely and thus for

all x ∈ X also Px-almost surely, since we can let tend h · m tend to δx when utilizing

the Feller property of Pt. Levy’s characterization of Brownian Motion then yields the

claim.

Lemma 5.5. Let the (E , D(E)) and Ξ be as in lemma 5.4 and f in D(E). Then for the

CAF of zero energy A[f ] belonging to (f(Ξt)− f(Ξ0))t≥0 one has

Eh·m(g(ΞS)A[f ]S ) = −

S∫0

E(PthPS−tg, f)dt ∀h, g ∈ L∞(X,m) ∩D(E),

where Pt is the semigroup generated by (E , D(E)) and ζ is a quasi-continuous version of

ζ.

Proof. We proceed as in the proof of lemma 5.4 by using the Markov property and the

additivity of A[f ] to obtain for S > T + ∆

Eh·m(g(ΞS)(A[f ]T+∆ − A

[f ]T ) = EPT h·m((PS−(T+∆)g)(Ξ∆)A

[f ]∆ )).


For f = R1η with η ∈ L2 the right hand side equals

〈PTh,E.

(PS−(T+∆)g(Ξ∆)

∆∫0

f(Ξu) + η(Ξu)du)〉m

and from this representation and the continuity of g(Ξ) it is clear that

lim∆→0

1

∆Eh·m(g(ΞS)(A

[f ]T+∆ − A

[f ]T ) = 〈PTh · PS−Tg, f + η〉m

= −E(PTh · PS−Tg, f).

For general f ∈ D(E) one establishes this result by the usual approximation argument

(compare [FOT94], thm. 5.2.4.). As before we now can compute

Eh·m(g(ΞS)A[f ]S ) = lim

∆→0

b S∆c−1∑i=0

∆1

∆Eh·m(g(ΞS)(A

[f ](i+1)∆ − A

[f ]i∆))

= − lim∆→0

(b S∆c−1∑i=0

∆ E(Pi∆h · PS−i∆g, f) +O(∆)S)

= −S∫

0

E(Pth · PS−tg, f)dt.

Lemma 5.6. Let (X, d) be an n-dimensional locally (L1, 1)-volume regular Alexandrov

space with lower curvature bound k and (E , D(E)) the canonical intrinsic Dirichlet form

on L2(X,B(X),m), where m is the n-dimensional Hausdorff-measure. Then for each

p ∈ X the distance function ρp(.) = d(p, .) : X 7→ R has the energy measure µ〈ρp〉 = m.

Proof. From the fact that Er → E for r → 0 pointwise on the set of Lipschitz functions

on (X, d), which serves as a common core for the forms Er and E , it follows that

µr〈f〉→µ〈f〉 for r → 0 weakly in the sense of Radon measures.

Now obviously

µr〈f〉(dx) =

∫B∗

r (x)

(f(x)− f(y)

d(x, y)

)2

mr(dy)mr(dx)

and in the special case f = dp the first variation formula for the distance function on

Alexandrov spaces ([OS94], thm 3.5) says that for fixed p, x ∈ X any choice of segments

γxy for y ∈ X the formula

ρp(x)− ρp(y) = d(x, y) cos infγpx

^pxy + ox(d(x, y))

obtains, where the infimum is taken over all possible choices of segments γpx connecting

p with x. Furthermore, for fixed p the cut locus Cp has measure zero (ibid., prop. 3.1),


which implies that ρp is differentiable in m-a.e. x ∈ X. Moreover, using Σx ' Rd for

m-a.e. x ∈ X, the volume regularity and the weak Riemannian structure of (X, d) we

find that

limr→0

∫B∗

r (x)

(f(x)−f(y)

d(x,y)

)2

mr(dy)1√

m(Br(x))=∫–

B1(0x)

〈γ′xp, z〉2gxdz = 1 for m-a.e. x ∈ X,

which yields the claim by Lebesque’s dominated convergence theorem.

Proposition 5.2. Let Ξ be the diffusion proceess generated by the canonical intrinsic

Dirichlet Form (E , D(E)) on an n-dimensional locally (L1, 1)-volume regular Alexandrov

space with lower curvature bound k. Then for any p ∈ X the process ρp(Ξ) satisfies the

stochastic differential inequality

dρp(Ξ) ≤ dBt + (n− 1)(lnSk)′ ρp(Ξ)dt Px-a.s. (5.18)

for all x ∈ X, where Bt is a real-valued standard Px-Brownian Motion.

Proof. Due to corollary 5.3 and lemma 5.6 we know that the MCAF-part in the Ito

decomposition of ρp(Ξ) is a real-valued Brownian motion. As for the CAF part A[ρp] of zero

energy we apply lemma 5.5 to f = ρp for arbitrary nonnegative h, ζ ∈ L∞(X,m) ∩D(E)

and the weak laplacian comparison (thm. 5.2), which gives

Eh·m(ζ(Ξs)A[f ]s ) = −

s∫0

E(Pth · Ps−tζ, ρp)dt

≤ (n− 1)

s∫0

〈Pth · Ps−tζ, (lnSk)′ ρp〉mdt

= (n− 1)Eh·m

ζ(Ξs)

s∫0

(lnSk)′ ρp(Ξt)dt

.Letting tend h · m to δx and using the monotone class theorem this means that A

[f ]s ≤

(n− 1)∫ s

0(lnSk)

′ ρp(Ξt)dt Px-a.s. and thus the proof is complete.

Since the radial process of a Brownian Motion on the Model space satisfies (5.18) with

”≤” replaced by ”=” one immediately obtains the

Theorem 5.5 (Brownian Motion Comparison Principle). Brownian Motion on X

is slower than on Mn,k in the following sense: for any x ∈ X let Ξx be the canonical

diffusion process on X starting in x then

E [ρx(Ξx,t)] ≤ E[ρx(Ξx,t)

]∀t ≥ 0,

where Ξx is Brownian Motion on Mn,k starting in some x ∈ Mn,k and ρx is the distance

function of x on Mn,k.

Appendix 71

6 Appendix

6.1 A - Remark on Coupling by Dirichlet Forms

In the Euclidean situation the coupling process for Brownian motion can also be con-

structed via a suitable Dirichlet form: the coupling process is characterized by its gener-

ator LRd

c on C2(Rd × Rd; R)

(LRd

c F )(x, y) = 〈A(x, y) : HessF (x, y)〉

with coefficients A = (αij) given by

(αij)(x, y) =1

2

(1Rd 1Rd − 2exy ⊗ exy

1Rd − 2exy ⊗ exy 1Rd

)and exy =

x− y

‖x− y‖

(cf. [LR86]), for which one obtains the decomposition (αij)(x, y) = σ(x, y)σ(x, y)t with

σ(x, y) =

(I −II I

)(Mxy

Mxy

)0

1 ...1

10 ...

0

=

(0 b2 ... bd b1 0 ... 0

0 b2 ... bd −b1 0 ... 0

)

where Mxy = (b1, b2, . . . , bd)(x, y) is a matrix of orthonormal basis vectors of Rd depending

smoothly on (x, y) ∈ Rd × Rd \ x = y such that b1(x, y) = exy. Obviously Mxy cannot

be extended smoothly across x = y. So the coupling operator on Rd×Rd is degenerate

(i. e. not strongly elliptic) and has discontinuities on the diagonal x = y. In order to

apply the theory of Dirichlet forms for the construction of the coupling process one needs

to find a measure m on Rd × Rd such that Lc is m-symmetric. If m = eβ(x)dx then∫Rd×Rd

φLcψ · dm =

∫Rd×Rd

φ(x, y) 〈A(x, y) : Hessψ(x, y)〉 eβ(x,y)dxdy

!=

∫Rd×Rd

〈∇φ,A∇ψ〉Rd×Rd dm ∀φ, ψ ∈ C∞0 (Rd × Rd)

is equivalent to

divA+ A · ∇β = 0

in the weak sense, where ( divA)i = αij,j. In the case given above one computes

divA(x, y) =(d− 2)

‖x− y‖2

((x− y)

(y − x)

)= A · v with v =

(d− 2)

‖x− y‖2

((x− y)

(y − x)

)such that one obtains as one particular solution to the previous equation

β(x, y) = −(d− 2) ln ‖x− y‖.

72 Appendix

Consequently one defines the coupling pre-Dirichlet form on L2(R2d, 1‖x−y‖(d−2)dx⊗ dy) by

Ec(u, v) =1

2

∫R2d

[〈∇xu,∇xv〉+ 〈∇yu,∇yv〉+ 〈∇xu,∇yv〉+ 〈∇yu,∇xv〉

−2(〈∇xu, exy〉〈∇yv, exy〉+ 〈∇yu, exy〉〈∇xv, exy〉)] 1

‖x− y‖(d−2)dxdy

for all u, v ∈ C∞0 (R2d) such that Ec(u, v) < ∞. Note that even if the generator of Ec can

be considered as an extension of the Laplacian on each component Rd this is not true for

the corresponding Dirichlet forms.

On a general Riemannian manifold it is much harder to find a symmetrizing measure for

the coupling operator Lc which is determined by a choice of Φ(., .) as in section 2.1. From

equation (2.4) one obtains the representation of Lc on smooth functions F : M ×M → R

LcF (x, y) =1

2〈ΣΦ : HessF 〉T 2

(x,y)(M×M) (x, y)

where one defines the bilinear form ΣΦ by

ΣΦ((Ux1 , V

y1 ), (Ux

2 , Vy2 )) =

⟨Φ−1(x, y)(Ux

1 , Vy1 ),(

1Rd 1Rd

1Rd 1Rd

)Φ−1(x, y)(Ux

2 , Vy2 )⟩

Rd×Rd.

As before the symmetry of Lc with respect to the volume measure with exponential density

eβ(x,y)m(dx)⊗m(dy) on M ×M is guaranteed by the condition

div(ΣΦ) + AΣΦ· ∇β !

= 0 (6.1)

where we used the identification T (p,r)(M × M) ∼= T p+r(M × M) : AΣ∼= Σ and the

definition div T =∑

i(∇eiT )(ei, .) for the divergence for a (1, r)-Tensor T with some

orthonormal basis ei, i = 1, . . . , 2d of T (M ×M). Now let us assume that Φ satisfies

conditions (∗) and (∗∗) of section 2, i.e. setting Syx = Φ1 Φ−12 (x, y) : Ty(M) → Tx(M)

and Sxy = Φ2 Φ−11 (x, y) accordingly, the requirement is

Sxy = Txy Rxy

where Txy is parallel translation of TxM to TyM along the curve γxy and Rxy is the linear

isometry on Tx by reflecting the γxy-direction. Then we obtain the representation for ΣΦ

ΣΦ((Ux1 , V

y1 ), (Ux

2 , Vy2 )) =

⟨(Ux

1 , Vy1 ),(

1TxM TyxRyx

TxyRxy 1TyM

)(Ux

2 , Vy2 )⟩

TxM×TyM

=⟨(Ux

1 , Vy1 ),(

1TxM Tyx

Txy 1TyM

)+ 2

(γxy⊗γyx

γyx⊗γxy

)(Ux

2 , Vy2 )⟩

TxM×TyM,

i.e. ΣΦ = ΣT + 2ΣΓ. Computation shows that

div ΣΓ(x, y) =(

(∆My dx(y)−1/d(x,y))γxy

(∆Mx dy(x)−1/d(x,y))γyx

)div ΣT (x, y) =

(divyTyx

divxTxy

).

Appendix 73

For the computation of divyTyx fix some normal basis ei of TyM , i.e. ∇eiej = 0 and

〈ei, ej〉 = δij. Then for some V ∈ TxM

〈divyTyx, V 〉x =∑

i

ei(〈Tyxei, V 〉x) =∑

i

∂

∂s |s=0〈V, Tσi(s)xσi(s)〉x

where σi is a (w.l.o.g. geodesic) integral curve of ei near y

=∑

i

∂

∂s | s=0t=1

〈Vi(s, t), //ci(s,.)(0,t) (σi(s))〉x

where c(s, t) : [−ε, ε[×[0, 1] → M is the geodesic variation of ci(0, .) = γyx(.) with fixed

end point x induced from ci(., o) = σi(.) and Vi(., .) is the vector field over ci(., .) obtained

by transporting V parallely along ci(s, t) first with respect to the t and then with respect

to s. Hence the summands equal

∂

∂s |s=0

1∫0

∂

∂t〈· · · , · · · 〉dt− ∂

∂s | s=0t=0

〈Vi(s, t), //ci(s,.)(0,t) (σi(s))〉x

where the second term vanishes, since V (., 0) is parallel and σi was chosen to be a geodesic.

=

1∫0

〈∇c∗∂∂s

∇c∗∂∂t

Vi(s, t), //ci(s,.)(0,t) (σi(s))〉x|s=0dt

+

1∫0

〈∇c∗∂∂t

Vi(s, t),∇c∗∂∂s

//ci(s,.)(0,t) (σi(s))〉x|s=0dt

where the second term vanishes, again since V (., 0) is parallel

=

1∫0

〈∇c∗∂∂t

∇c∗∂∂s

Vi(s, t), //ci(s,.)(0,t) (σi(s))〉x|s=0dt

+

1∫0

〈R(c∗∂

∂t, c∗

∂

∂s)V (s, t), //

ci(s,.)(0,t) (σi(s))〉x|s=0dt

where the first integral is zero because ∇c∗∂∂s

V (s, t) = 0

=

1∫0

〈R(γyx(t), Ji(t))//γxy

(0,1−t)V,//γyx

(0,t)ei〉xdt

= −⟨ 1∫

0

//γyx

(t,1)

R(γyx(t), Ji(t))//

γyx

(0,t)ei

dt, V

⟩x

74 Appendix

with the Jacobi-field Ji along γyx obtained from the geodesic variation ci(., .).

Hence the the equation (6.1) for ∇β = (∇xβ,∇yβ) becomes(∇xβ + Tyx∇yβ + 2γxy〈γyx,∇yβ〉∇yβ + Txy∇xβ + 2γyx〈γxy,∇xβ〉

)!= −

(2(∆M

y dx(y)− 1/d(x, y))γxy + divyTyx

2(∆Mx dy(x)− 1/d(x, y))γyx + divxTxy

),

and hence we are left with the question under which conditions on (M, g) this equation

for β is solvable.

Conversely one might try the analogue of the the Euclidean coupling form in the manifold

case by defining

Ec(u, v) =1

2

∫M×M

[〈∇xu,∇xv〉x + 〈∇yu,∇yv〉y + 〈∇xu, Tyx∇yv〉x + 〈∇yu, Tyx∇xv〉y

+2(〈∇xu, γxy〉x〈∇yv, γyx〉y + 〈∇yu, γyx〉y〈∇xv, γxy〉x)]eβ(x,y)m(dx)m(dy)

or more general

Ec(u, v) =1

2

∫M×M

(∇xu

∇yu

)(a11 a12

a21 a22

)(∇xv

∇yv

)eβ(x,y)m(dx)m(dy)

where aij = aij(x, y) are linear maps between the corresponding tangent spaces, such that

the associated operators on functions of the form u = f ⊗ 1 (⇔ ∇yu = 0) is

Lc(f ⊗ 1)(x, y) = 〈a11,Hess f〉x + 〈∇xf, divxa11 + a11∇xβ + divya12 + a12∇yβ〉x. (6.2)

If Ec is to generate a coupling X ∈ M ×M between the marginal processes X1 = π1X

and X2 = π2X then the corresponding condition for the generators Lc, L1 and L2 reads

as follows

Lc(f ⊗ 1) = (L1f)⊗ 1 and Lc(1⊗ g) = 1⊗ (L2g) (6.3)

(equivalently Lc(f π1) = (L1f) π1 and Lc(g π2) = (L2g) π2) which leads to the

requirement that the operator in equation (6.2) should not depend on y, i.e.

a11(x, y)!= a11(x) and V (x, y) := a11∇xβ + divya12 + a12∇yβ

!= V (x)

that is, if we have fixed a diffusion process on each factor of M×M with generator L1 and

L2 respectively then the condition (6.3) gives first order equations for some appropriate

choice of aij and β in terms of the data L1 and L2.

Appendix 75

6.2 B - More about the Geometry of Alexandrov Spaces

Rough Dimension and Natural Coordinates

Definition 6.1. The rough dimension of a bounded subset V ⊂ X in a metric space is

defined as

dimr V = infα > 0

∣∣∣ limε→0

εαβV (ε) = 0

where βV (ε) is the cardinality of a maximal ε-net in V , i.e. the cardinality of a maximal

subset xi | i ∈ I ⊂ V such that d(xi, xj) ≥ ε.

Note that the rough dimension is an upper bound for the Hausdorff dimension. If (X, d) is

an Alexandrov space the convexity of geodesic triangles implies that the rough dimension

is locally constant. This follows from the observation that for fixed p, q ∈ X with d(p, q) ≤δ and a given ε-net xi ⊂ Vp of a neighborhood Vp 3 p the image points x′i of xiunder a geodesic contraction with respect to q will be an ε′-net of Vq where Vq is some

appropriately chosen neighbourhood of q and where 0 < ε′ → 0 for δ → 0. This argument

can be made rigorous to yield the estimate dimr Vp ≤ dimr Vq for q sufficiently close to p

and Vp and Vq sufficiently small. Thus n = dimr X = limVp→p dimr Vp is well defined,

i.e. independent of p.

Definition 6.2. For p ∈ X a system of tuples (a−i, a+i | i = 1, . . . ,m is called a (m, δ)-

strainer at p iff

âipaj ≥

π/2− δ if i 6= ±jπ − δ if i = −j.

In this case the map

φai : X ⊃ Vp → Rm, q → (d(a1, q), . . . , d(am, q))

is called the natural coordinate function with base ai | i = 1, . . . ,m.

The idea about a strainer in a point p is that if p admits a system of tuples (a−i, a+i) ∈X ×X | i = 1, . . . ,m such that the corresponding broken geodesic segments γa−ip ∗ γpa+i

are almost geodesic in p and are also almost mutually perpendicular then the endpoints

ai naturally define a coordinate system on X in a neighborhood of p, provided m is

the maximal number of such points for given p. In fact, if (a−i, a+i) | i = 1, . . . ,m is

an (m, δ)-strainer at p with δ ≤ 1/2m and such that there is no (m+ 1, 4δ)-strainer at p

then φai provides a bi-Lipschitz homeomorphism of some neighborhood Up of p onto an

open subset of Rm (equipped with the metric stemming from the maximum-norm), see

thm. 5.4. in [BGP92]. Since in that case the Hausdorff-dimension, the topological and

the rough dimension of Up are equal to n, the maximal strain index Inds(p) at p is, in

fact, constant on X, which is used for the definition of dim(X).

76 Appendix

Weak Riemannian Structure ([OS94])

For a given (n, δ)-strainer (a−i, a+i) at p ∈ X and sufficiently small δ let Uai be some

neighborhood of p which is mapped (bi-Lip) homeomorphically onto an open set in Rn

([BGP92]), the set Vai = Uai∩⋂

i=1,...,nCaiwith Cai

= y ∈ X |∃!γyai is the set where

all geodesic segments from a point in x ∈ Uai to the base points ai are unique. Then

the function

gai : Uai → Rn×nsym , x→ (cos ^(γxai

, γxaj))ij

is uniquely defined and continuous on Vai ([OS94]). Starting from this observation one

can show that the systemφai, gai, Uai, Vai

(a−i,a+i) is a (n,δ)-strainer at p, p∈X\SX ,δ≤1/2n

is weak C1-Riemannian structure on X \SX ⊂ X, i.e. it possesses the following properties

- for all ai, Vai ⊂ Uai, Uai ⊂ X open.

- every φai maps Uai homeomorphically onto an open set in Rn

- Vai is a covering of X \ SX .

- if two maps φai and φbi satisfy Vai ∩ Vbi 6= ∅ then φai φ−1bi is differentiable

on φbi(Vai ∩ Vbi ∩ (X \ SX)

)- for each ai the map gai φ−1

ai is continuous on φai(Vai ∩ (X \ SX)

)- for any x ∈ Vai ∩ Vbi ∩X \ SX

gbi(x) =d

dx

(φai φ−1

bi)t

(φbi(x)) · gai(x) ·d

dx

(φai φ−1

bi)(φbi(x)).

Thus the difference between a Riemannian and a weak Riemannian structure is basically

that the topological types of X \ SX and Vai are undetermined. In particular X \ SX

need not be a manifold.

Excess measure on Alexandrov Surfaces ([Ale55, Mac98])

The excess of a geodesic triangle ∆ in an Alexandrov space (X, d) is defined by

e(∆) =3∑

i=1

înti (∆)− π

where înti (∆) denote the interior angles of ∆. The following observation is the basis for

the definition of the excess measure for general subsets in X. Let D be an open polygonal

Appendix 77

domain in a two-dimensional Alexandrov space, i.e. the boundary ∂D is given by a set

of geodesic curves γi, i = 1, . . . , I, and let ∆j, j = 1, . . . , J be a decomposition of D into

geodesic triangles, then

J∑j=1

e(∆j) +K∑

k=1

(2π − L(Σpk)) = 2πχ(D)−

I∑i=1

(π − βi)

where pk, k = 1, . . . K are the vertices of the decomposition D =⋃

∆j lying in (the

interior of) D and L(Σpk) denotes the length of their space of directions (dim Σpk

= 1),

χ(D) is the Euler-Poincare-characteristic of D in X and βi, i = 1, . . . , I denotes the

interior angle between the geodesics γi−1 and γi. Since in this equality the right hand side

does not depend on the decomposition ∆j the total excess of D is defined by

e(D) =J∑

j=1

e(∆j) +K∑

k=1

(2π − L(Σpk))

This defines an finitely additive measure on the class D of subsets of X which can be

represented as finite unions or set-theoretic differences of geodesic triangles, segments

and points. By the usual procedure e(.) is extended as an outer measure on X which

is countably additive on the σ-algebra generated by D. Finally, by choosing the ∆j

sufficiently small and by comparing them with the almost congruent geodesic triangles in

R2 obtained from the natural coordinate functions φai of the previous paragraphs one

can deduce the central estimate (cf. [Mac98])

e(D) ≥ κm(D) ∀D ∈ D

which implies in particular that e(.)+ |κ|m(.) extends to a positive outer measure on X.

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