Intrinsic Coupling on Riemannian Manifolds and Polyhedra

Starting from a central limit theorem for geometric random walks we give an elementary construction of couplings between Brownian motions on Riemannian manifolds. This approach shows that cut locus phenomena are indeed inessential for Kendall's and Cranston's stochastic proof of gradient estimates for harmonic functions on Riemannian manifolds with lower curvature bounds. Moreover, since the method is based on an asymptotic quadruple inequality and a central limit theorem only it may be extended to certain non smooth spaces which we illustrate by the example of Riemannian polyhedra. Here we also recover the classical heat kernel gradient estimate which is well known from the smooth setting.

1. Introduction. The Riemannian coupling by reflection technique by Kendall [Ken86] was used by Cranston [Cra91] to give an elegant stochastic proof of L ∞ -gradient estimates for harmonic functions on Riemannian manifolds with lower Ricci curvature bounds. The stronger pointwise bounds due to Yau [Yau75] are usually proved by analytic arguments based on the fundamental Bochner-Lichnerowicz formula which is difficult to transfer to non smooth (cf. [CH98]) or even non Riemannian situations. Therefore one may turn to the more flexible stochastic methods for the analysis of second order differential operators on non smooth or metric measure spaces (X, d, m) in terms of their associated Markov processes. This general agenda constitutes the background motivation for the present attempt to simplify Cranston's stochastic proof of gradient estimates whose beautiful geometric content is hidden behind a sophisticated and, as it turns out, dispensable technical superstructure. The basic idea behind the coupling by reflection method on Riemannian manifolds (M, g) is to construct a stochastic process Ξ on the product M × M such that i) each factor Ξ 1 = π 1 (Ξ) and Ξ 2 = π 2 (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 T N (ξ) can be estimated from above.
The standard construction of Ξ put forth in [Ken86] uses SDE theory on M × M and hence requires a certain degree of smoothness on the coefficients. Unfortunately these coefficients typically become singular on the diagonal and, more severely, on the cut locus Cut(M ) ⊂ M × M . Even if cut locus phenomena in geometric stochastic analysis have been addressed occasionally (cf. [CKM93,Wan94,MS96]) they remain a delicate issue especially for Kendall's coupling method. We overcome these difficulties by introducing a more intrinsic construction of the coupling process on M ×M which yields the essential coupling probability estimate irrespective if the manifold (M, g) is Cartan-Hadamard or not. Moreover, a brief analysis of the proofs reveals that except an asymptotic quadruple inequality for geodesics in (M, g) and a central limit property for random walks no further regularity of (M, g) seen as a metric measure space (X, d, m) is required. Hence, the method may be applied in more general situations as we indicate by the example of certain Riemannian polyhedra. Finally we point out that the simpler coupling by parallel transport method works well also in the polyhedron case from which we deduce gradient estimates for the corresponding heat semigroup.
Acknowledgments: To Karl-Theodor Sturm for suggesting the problem.

Preliminaries.
2.1. Riemannian Central Limit Theorem [Jør75]. Let (M, g) be a smooth Riemannian manifold of dimension n and fix for every x ∈ M an isometry Φ x : R n → T x M such that the resulting function is measurable. Let (ξ k ) k∈N be a sequence of R n -valued and independent random variables defined on some probability space (Ω, O, P ) whose distribution equals the normalized uniform distribution on S n−1 . A geodesic random walk (Ξ ,x k ) k∈N with step size > 0 and starting point x ∈ M is given inductively by , where exp is the exponential map of (M, g). Then one may consider the sequence of subordinated processΞ x k (t) := Ξ for k ∈ N 0 , where τ k is a Poisson jump process on N with parameter k. For k fixed Ξ x k (t) is a (time homogeneous) Markov process with transition function with µ (z, A) = -S n−1 z ⊂TzM 1 A (exp z ( θ))dθ and µ i = µ • µ · · · • µ. The generator ofΞ k , or equivalently of the semigroup (P k t ) t≥0 , is therefore given by where Af (x) = 1/2n∆f (x) with the Laplace-Beltrami operator ∆f (x) = trace(Hess f )(x) on (M, g), see [Blu84,vR02]. Using (1) and Kurtz' semigroup approximation theorem it is easy to show thatP k t −→P t for k −→ ∞ in the strong operator sense where P t = e tA = P M t/2n is the heat semigroup P M t = e t∆ on (M, g) after a linear time change t → t/2n. Thus weak convergence for the familỹ Ξ k · to a (time changed) Brownian motion Ξ x · starting in x is obtained from standard arguments showing that the sequence of distributionsΞ k · is tight on the Skorokhod path space D R+ (M ).
2.2. Coupled Random Walks. Instead of using the SDE approach to the construction of a coupling of two Brownian motions on (M, g) we follow the lines of the Markov chain approximation scheme for solutions to martingale problems for degenerate diffusion operators in the sense of [SV79] where the approximating processes are coupled geodesic random walks on M × M .
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 the set Cut(M ) ⊂ M × M is defined as the collection 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. Proof. Suppose first that we found a measurable symmetric function γ : M × M → C 1 ([0, 1], M ) as above and let ψ i ∈ 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 T x M ⊕T y M by Φ(x, y) = {γ xy / γ xy ,ψ 2 1 , . . . ,ψ n 1 ,γ yx / γ yx ,ψ 2 2 , . . . ,ψ n 2 } out of the frame {ψ 1 (x), ψ 2 (y)} via Schmidt's orthogonalization procedure applied to the vectors ψ i (e k ), k = 1, . . . , n in the orthogonal complements ofγ xy andγ yx in T x M and T y M respectively. Since the maps ∂ s |s=0 and ∂ s |s=1 : C 1 ([0, 1], M ) → T M are continuous and the construction of the basis Φ in T x M ⊕ T y M depends continuously on the data {ψ 1 , ψ 2 },γ xy and γ yx 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 R l and some complete ordering on R l ) and restrict the discussion to the closed subset We define a measurable set-valued map Γ : D − (M ) → 2 C 1 ([0,1],M ) as follows: for each > 0 choose some -net P = {p i | i ∈ N} in D − (M ) and choose some minimal geodesic γ p i ,p j for each pair of points p i , p j ∈ P . Arrange the set of pairs (p i , p j ) into a common sequence {(p i k , p j k ) | k ∈ N} and let γ : It is clear from the definition that the functions γ 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 {γ xy } >0 are relatively compact in C 1 ([0, 1], M ). Trivially any limit point of {γ xy } >0 for tending to zero will be a minimal geodesic from x to y. Let us choose a priori some sequence k → 0 for k → ∞ then we define the set Lemma 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 f k : X → Y be a sequence of measurable functions which are pointwise relatively compact, i.e. for all x in X the set {f k (x)} k∈N is relatively compact in Y . Let the set valued map F : X → 2 Y be defined by pointwise collecting all possible limit points of the sequence f k . Then there is a measurable 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 Since any open O ⊂ Y can be exhausted by countably many set of the type B δ (y) with δ > 0, y ∈ Y we may replace O by B δ (y) in the condition above. But using the pointwise compactness of the sequence f k and a diagonal sequence argument it is easy to show that Choosing some sequence δ l δ we see that in fact F −1 B δ (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 (ξ k ) k∈N and (η k ) k∈N of R n -valued i.i.d. random variables with normalized uniform distribution on S n−1 and define a coupled geodesic random walk Ξ where π i , i = 1, 2 are the projections of M × M on the first and second factor respectively. We have two canonical possibilities to extend Ξ ,(x,y) k to a process with continuous time parameter t ∈ R + , namely i) by geodesic interpolationΞ In particular, choosing = 1/ √ k and λ = k in ii) for k ∈ N one obtains a sequence of Markov processesΞ where the kernel µ : M 2 × B(M 2 ) → R is given by The generator of the semigroup (P t k,(x,y) ) t≥0 induced byΞ k,(x,y) is 3. Coupling Central Limit Theorem. The following lemma is a partial but for our aim sufficient characterization of any limit of the sequence of operators (L k ) k∈N .
Lemma 3. Let F : M × M → R be a smooth function. Then with A = 1/2n ∆ M and the bilinear form ., . Φ(x,y) : and the definition of the exponential map yield due to the smoothness of the data (M, g) and the function F . Inserting this into the definition of L k gives is tight on D R+ (M × M ) and any weak limit of a converging subsequence (Ξ k ,(x,y) · ) k is a solution to the martingale problem for L c in the following restricted sense: let denote the canonical process on M ×M induced from a limit measure w- x,y martingale with initial value 0. In particular, under P ∞ x,y both marginal processes (Ξ 1 s := π 1 s ) s≥0 and (Ξ 2 s := π 2 s ) s≥0 are Brownian motions on (M, g) starting in x and y respectively.
Any probability measure on D R+ (M × M ) with the properties above is called a solution to the (restricted) coupling martingale problem. We do not claim uniqueness nor a Markov property. Note that we circumvented the problem of the cut locus by the choice of admissible test functions F .
Lemma 4 (Tightness criterion on D R+ (X)). Let (X, d) be a complete and separable metric space and let (Ω l , P l , (Ξ l t ) t≥0 ) l∈N be a sequence of cadlag processes on X. Then the following condition is sufficient for tightness of (Ξ l · ) * (P l ) on D R+ (X): For all N ∈ N, and η, > 0 there are x 0 ∈ X, l 0 ∈ N and M, δ > 0 such that . We omit the details which can be found in [vR02] and turn to the martingale problem for L c . Since L k generates the process Ξ k,(x,y) · this is also true for its realization (D R+ (M × M ), (Ξ k,(x,y) · ) * P, (π s ) s≥0 ) on the path space and which is in this case equivalent to 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-lim k →∞ (Ξ k ,(x,y) · ) * P exists. Finally, the assertion concerning the marginal processes follows either from [Jør75] or from putting F = 1⊗f and F = f ⊗1 in (6) 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 estimate it is easier to work with the continuous interpolated processes (Ξ k,(x,y) · ) k as approximation of a suitable limit Ξ (x,y) · . Therefore we need the following ) * P is, in which case the limits coincide. In particular the family ((Ξ k,(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 C R+ (M × M ).
Proof. By construction of (Ξ ) has continuous paths, so that for every possible weak limit of a converging subsequence (Ξ k ,(x,y) · ) * P and since the sequence Θ k converges to Id R+ weakly, the continuity argument in section 17. of [Bil68] can be applied, giving for that specific subsequence, i.e. we have shown that for any subsequence k → ∞ To prove the other implication note first that the trivial estimate implies the almost sure continuity of the coordinate process π · w.r.t. any weak limit of (Ξ k,(x,y) · ) * P . We may also write and thus we may argue as before. Finally, the compactness itself comes from theorem 1 as well as as the characterization of any limit as a solution to the (restricted) coupling martingale problem via equation (7).

Coupling Probability and Gradient
Estimates for Harmonic Functions. 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.
Note that in the statement above the emphasis lies on the fact that the estimates (8) and (9) remain true also in Cut(M ).
Proof. The proof can be found in any textbook on Riemannian geometry as long as y is not conjugate to x along γ and is based on the second variation formula for the arc length functional. In the case that y is conjugate to x along γ one may show (8) by subpartitioning γ = γ 1 * γ 2 * · · · * γ k into geodesic segments {γ i } without conjugate points. By triangle inequality the individual estimates (8 i ) along {γ i } can be reassembled to yield (8) along γ xy . The proof also shows that the error term o γ (t 2 ) in (8) may be replaced by a uniform error estimate o(t 2 ) as long as x = y range over a compact K ⊂ M × M \ D(M ). For the remaining few details the reader is referred to [vR02].
In order to apply the previous lemma to the the sequence (Ξ l,(x,y) . ) l we require in addition to condition ( * ) on page 11 that Φ(., .) satisfies Theorem 2 (Coupling Probability Estimate [Ken86]). Let Φ be chosen as above and Sec(M ) ≥ −κ < 0. Then for arbitrary x, y ∈ M and any weak limit P ∞ x,y = w-lim l →∞ (Ξ l ,(x,y) · ) * P on C R+ (M × M ) the following estimate holds true: Proof. For x = y there is obviously nothing to prove. So let (x, y) ∈ D(M ) and assume first that M is compact. For δ > 0 let T l D,δ : we obtain from lemma 5 where λ = pr 1 θ is the projection of θ onto the first coordinate axis and χ = θ ⊥ 2 R n is the squared length of the orthogonal part of θ. This estimate inserted into the inductive definition of (Ξ l,(x,y) t ) t≥0 yields in the caseΞ with the random variables λ i = pr 1 ξ i and χ i = ξ ⊥ i 2 R n , from which one deduces by iteration at least on the set {T l D,δ > t}, with Define furthermore the stopping times The last equality is a consequence of Donsker's invariance principle applied to the sequence of processes (r l · ) l∈N : since each (λ i ) i and (χ i ) i are independent sequences of i.i.d. random variables on {Ω, P, A} with one finds that (r l · ) l converges weakly to the process r ∞ · with such that in particular P P,Ω ({T δ (r ∞ ) = m}) = 0 and we can pass to the limit for l → ∞ in the last term on the right hand side of (11). Letting m tend to infinity leads to where δ > 0 was chosen arbitrarily from which we finally may conclude with T 0 being the first hitting time of the origin for the semi-martingale r ∞ . Using a Girsanov transformation of (Ω, O, P ) the probability on the right hand side can be computed precisely to be which is the claim in the compact case. For noncompact M we choose some open precompact A ⊂ M such that (x, y) in K. We may stop the processes (Ξ l ,(x,y) · ) when they leave A and repeat the previous arguments for the the stopping time T A,D,δ = T D,δ ∧ T A c with T A c = inf{s ≥ 0 | ω s ∈ A c } which gives instead of (11) From this we obtain (13) if we successively let tend l → ∞, A → M × M , δ → 0 and n → ∞.
Also in the case of lower Ricci curvature bounds the same type of arguments should yield the extension of theorem 2. However, the difficulties arise from the fact that lower Ricci bounds lead to a uniform upper estimate of the expectation of χ i in (10) only. Since these random variables are also only asymptotically mutually independent, one has to find and apply an appropriate central limit theorem to the expression 1 l lt i=0 χ i+1 in order to obtain the pathwise(!) upper bound for the distance process by the semimartingale (12).
For different (local or global) versions of the following result as well as for extensions to harmonic maps the reader is referred in particular to the works by W. Kendall, M. Cranston or more recently by F.Y. Wang.
Corollary 2 (Gradient estimate [Cra91]). If u is a harmonic, nonnegative and uniformly bounded function on M , then Proof. From elliptic regularity theory we now that u ∈ C ∞ (M ). Let x = y be given. Since ∆u = 0 we find L c (u⊗1) = L c (1⊗u) = 0 and from theorem 1 it follows that both processes ((u ⊗ 1)(π s )) s and ((1 ⊗ u)(π s )) s are nonnegative continuous bounded martingales with respect to the probability measure P ∞ x,y , where π · = (π 1 , π 2 ) · is the projection process on the path space C R+ (M × M ). For any s > 0 we obtain by means of the optional stopping theorem and finally, with sup x,y |u(x) − u(y)| ≤ u ∞ following from u ≥ 0 ≤ u ∞ P P ∞ x,y ({T D ≥ s}) Passing to the limit for s → ∞ proves the claim by theorem 2.
3.2. Coupling by Parallel Transport and Heat Kernel Gradient Estimate. Instead of the coupling by reflection one may also consider coupling by parallel transport when conditions ( * ) and ( * * ) on Φ are replaced by Due to (8) this leads to the estimate for any limiting measure P ∞ x,y on C R+ (M × M ). Since any such P ∞ x,y is a coupling for (M, g)-Brownian motions starting in x and y this implies a gradient estimate for the heat semigroup P t = e ∆t on (M, g) of the form for all f ∈ C 1 c (M ) and x ∈ M , cf. [Wan97] as well as for applications.
4. Extension to Riemannian Polyhedra. Let X be an n-dimensional topological manifold equipped with a complete metric d. We call (X, d) an n-dimensional Riemannian polyhedron with lower curvature bound κ ∈ R if (X, d) is isometric to locally finite polyhedron i P i of convex closed patches P i ⊂ M n i (i ∈ I) of n-dimensional Riemannian manifolds with uniform lower sectional curvature bound κ, where i) the boundary ∂P i = j S ij ⊂ M i of each patch P i ⊂ M i is the union of totally geodesic hypersurfaces S ij in M i ii) each S ij ⊂ X is contained in the intersection of at most two P k ⊂ X and S ij ⊂ M i where S kl ⊂ M k are isometric whenever two adjacent patches P i ⊂ X and P k ⊂ X have a common face S ij S kl ⊂ (X, d) iii) the sum of the dihedral angles for each face of codimension 2 is less or equal 2π.
Examples. The boundary ∂K of a convex Euclidean polyhedron K ⊂ R n (with nonempty interiorK) 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 : with c ∈ (a, b) and f (c) = 0 and f /f = −k 2 1 on [a, c) −k 2 4.1. Constructions. The 2π-condition iii) above assures that (X, d) is an Alexandrov space with Curv(X) ≥ κ (cf. [BBI01]), and we can use the result in [Pet98] that a geodesic segment connecting two arbitrary metrically regular points does not hit a metrically singular 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 X n−2 of X. Thus there is a natural parallel translation along any geodesic segment γ xy whenever x and y are regular and which is obtained piecewisely from the parallel translation on the Riemannian patches P i and from the natural gluing of the tangent half-spaces for points x ∈ X n−1 \ X n−2 lying on the intersection of two adjacent (n−1)-faces S ij S kl ⊂ X. Similarly we can define the exponential map exp x for every regular point x ∈ X, i.e. for given ξ ∈ T x X we obtain a unique 'quasi-geodesic' curve R + t → exp x (tξ) (and which can be represented as a union of geodesic segments on the patches P i ). With these constructions at our disposal we can verify a non-smooth version of the asymptotic quadrangle estimate of lemma 5: Theorem 3. Let (X, d) be a n-dimensional Riemannian polyhedron with lower curvature bound −κ < 0 and let x, y ∈ X \ X n−2 be connected by some segment γ xy . Then for ξ ∈ T x X, ξ = 1 the estimate (9) holds, where the error term o(t 2 ) can be chosen uniform if x = y range over a compact subset of X \ X n−2 .
Proof. Let us prove (9) for fixed x, y ∈ X \X n−2 and ξ ∈ T x X first, i.e. without addressing the problem of uniformity. Suppose furthermore that for some P i we have γ xy ⊂ P i , i.e. γ xy is entirely contained in the (closed) patch P i , then we distinguish three cases: i) If γ xy ⊂P i then due to lemma 5 there is nothing left to prove.
ii) x ∈P i and y ∈ P i ∩ P j for some j. Since y is assumed to be regular the first order part of estimate (9) 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 (9). If γ xy is orthogonal to the hypersurface ∂P i ∩ ∂P j at y or x and y are both in P i ∩ P j then again there is nothing to prove since in this case we have to consider geodesic variations which take place completely on one of the patches P i ⊂ M i or P j ⊂ M j and we can apply lemma 5 on M i or M j respectively. Consequently we only have to treat the case that γ xy is neither parallel nor orthogonal to ∂P i ∩ ∂P j , i.e. 0 < γxy(d(x,y)) γxy(d(x,y)) , ν Ty Mi < 1 where ν denotes the outward unit normal vector of ∂P i . Let η = / / γxy ξ be the parallel translate of a unit vector ξ ∈ T x M i normal toγ xy .
Then ζ = η − ν,η ν,γxy(d(x,y)) γ xy (d(x, y)) ∈ T y M i ∩ T y M j is the unique vector in the intersection of the {γ xy (d(x, y)), η}-plane and the tangent hyperplane to ∂P i in y which is determined by its w.r.t.γ xy orthogonal projection η. For its length we obtain ζ = sin −1 α where α is the angle enclosed byγ xy and η at y. Since ∂P i ⊂ M i and ∂P j ⊂ M j are totally geodesic the point z = exp y (tζ) also lies on ∂P i ∩ ∂P j and the triangle inequality yields where d X , d Mi and d Mj denote the distance functions on X, M i and M j respectively. Now the estimate (8) of lemma 5 applied to ξ and ζ in M i yields since trivially γ, ζ = cos α sin α and by construction ζ ⊥ = η = / / γxy ξ. As for the distance d Mj (z, exp y (tη)) remember that by the smoothness assumption the curvature of M j is locally uniformly bounded and from the Toponogov triangle comparison and the cosine formula on the model spaces M d,κ we may infer with β = TyMj (ζ, η) because all vectorsγ xy , 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 (15) yields (9).
iii) x ∈ P i ∩ P k and y ∈ P i ∩ P j . We may argue similarly as in ii) by subdividing the quadruple into two geodesic triangles on M j and M k and a remaining quadruple on M i . -Alternatively, if z ∈ γ xy ∩P i = ∅ then one may subdivide γ xy = γ xz * γ zy and argue as in ii). If γ xy ∩P i = ∅, then again we have to deal with variations of γ xy on a single Riemannian patch P k only, where P k depends on the direction ξ⊥γ xy .
The discussion above proves (9) when γ xy ⊂ P i for some P i . In the general case when γ xy is not contained in a single patch we subdivide γ xy = γ i1 * γ i2 * · · · * γ im into pieces γ i k ⊂ P i k lying entirely on one of the patches which we consider separately: let {x 1 , . . . , x m } = γ xy ∩ due to the special choice of the directions {z k |k ∈ 1, . . . , m}. Finally, the uniformity assertion is obtained in a similar way by combining the arguments of lemma 5 on each patch P i with the observation that for a given compact set K ⊂ X \ X n−2 the collection of all segments {γ xy | (x, y) ∈ K × K} also lies at a strictly positive distance away from X n−2 , which may be inferred from a simple compactness consideration. This implies that there is some t 0 > 0 such that for all t ≤ t 0 and x, y ∈ K all variations γ xy,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 as in lemma 5 that the estimate (9) 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 1. The ('Dirichlet-')Laplacian ∆ X on (X, d) is defined as the generator of the Dirichlet form (E, D(E)) which is obtained as the L 2 (X, dm = i dm i P i )-closure of the classical energy form E(f, f ) = i Pi |∇f | 2 dm i 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 (E, D(E)) on L 2 (X, dm) is called a Brownian motion on (X, d).
Equivalently we could define (E, D(E)) by the sum of the Dirichlet integrals on the patches P i as above with the domain D(E) equal to the set of piecewise H 1,2 (P i )-functions f ∈ L 2 (X, dm) with E(f, f ) < ∞ and whose traces f |S± along the joint (n−1)-dimensional faces S = P i ∩ P j of each pair of adjacent patches coincide.
For the construction of the coupling process on X × X for two Brownian Motions on (X, d) we would like to proceed as in the smooth case by using a coupling map Φ(., .) with Φ(x, y) : R n × R n → K x X × K y X where K x denotes the tangent cone of X over x (c.f. [BBI01]). Here further singularities of Φ(. , .) may be caused by the existence of non-Euclidean tangent cones K x when x ∈ X n−2 . However, choosing beforehand a map Ψ(, ) on X × X n−2 ∪ X n−2 × X ∪ {(x, x) | x ∈ X} with Ψ(x, y) : R n × R n → K x X × K y X (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 × X n−2 ∪ X n−2 × X) which can be proved by slightly modifying the arguments of lemma 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 (Ξ k,(x,y) · ) k and (Ξ k,(x,y) · ) k by scaling.
Proof. The tightness assertion and the coincidence of the limits of any jointly converging subsequences (Ξ k ,(x,y) · ) k and (Ξ k ,(x,y) · ) k is proved precisely in the same manner as in the smooth case. Let us denote for short µ := P ∞ x,y = w-lim k →∞ (Ξ k ,(x,y) · ) * P the weak limit of some converging subsequence. Then it remains to identify the marginals µ i = Π i * µ of µ under the projection map Π i : C R+ (X × X) → C R+ (X), ω. → ω i . as the measures induced from the Dirichlet form (E τ , D(E τ )) := ( 1 2n E, D(E)) and starting points x and y respectively. For ρ > 0 let C ρ be some Lipschitz ρ−neighbourhood of the set X n−2 and let (E τ ρ , D(E τ ρ )) be Dirichlet form which is obtained by taking the L 2 (X, dm)-closure of the energy form E τ restricted to the set of Lipschitz functions with compact support in X \ C ρ . Furthermore let T i ρ = inf{t ≥ 0|ω i t ∈ C ρ } be the hitting time of the marginals for the set C ρ . We claim that the T i ρ -stopped marginal processes under any weak limit P ∞ x,y are associated with (E τ ρ , D(E τ ρ )) starting in x and y respectively. For this denote by A ρ the collection of all f ∈ i C ∞ (P i ) ∩ Lip(X) ∩ C c (X \ C ρ ) satisfying the gluing condition for the normal derivatives on adjacent (n − 1)-dimensional faces Since for the generator L k ofΞ k,(x,y) · we have that with A k acting as a mean value type operator according to formula (1) for f ∈ A ρ it is easy to show (see also proof of lemma 6) that where ∆ X ρ is the restriction of the formal Laplace-Beltrami-operator ∆ X = i ∆ Pi |Pi onto the set A ρ . Trivially in the formula above we may replace t by t ∧ T i ρ .
Let us assume first that X is compact and, by abuse of notation, let us denote by ∆ X ρ also the generator of (E ρ , D(E ρ )), whose domain equals Then by using the existence and smoothness of the heat kernel (p X,ρ t (., .)) t≥0 associated to ∆ X ρ we see that for f ∈ D(∆ X ρ ) and > 0 and for → 0 where in the last line we used the fact that on D(∆ X ρ ) the heat semigroup (P X,ρ t ) commutes with ∆ X ρ . This implies that ( 1 2n ∆ X ρ , A ρ ) is a core for the generator ∆ X ρ of the Dirichlet form (E τ ρ , D(E τ ρ )) and hence by (16), abstractly speaking, we may say that the measures µ i ρ = (Π i • Σ T i ρ ) * µ induced from the projection Π i and the stopping maps Σ T i ρ : C R+ (X × X) → C R+ (X × X), ω . → ω .∧T i ρ (i = 1, 2) are solutions to the martingale problem for ( 1 2n ∆ X ρ , δ x ) and ( 1 2n ∆ X ρ , δ y ) respectively. Since these martingale problems are well-posed the measures µ i ρ must coincide with those generated by the Dirichlet form (E τ ρ , D(E τ ρ )) and given starting points x and y. In order to show that Π i * µ is induced from (E τ , D(E τ )) we want to pass to the limit for ρ → 0 in the last statement. It is easy to see that weak convergence (Σ ρ ) * µ −→ µ will follow from lim ρ→0 µ{T i ρ ≤ t} = 0 for all t > 0. Moreover, obviously for < ρ, where µ i is defined as above and T ρ is the first hitting time functional of C ρ on the path space C R+ (X). Remember that µ i is induced from the Dirichlet form (E τ , D(E τ )).
Using the fact that the set X n−2 is polar for (E, D(E)) it follows that for any subsequence → 0 the Dirichlet forms (E τ , D(E τ )) converges to (E τ , D(E τ )) in the sense of Mosco (cf. [Mos94]), which implies the L 2 (X, dm)−strong convergence of the semigroups P t to the semigroup P t generated by (E τ , D(E τ )). Hence, by standard compactness arguments and the fact that {T ρ ≤ t} ⊂ C R+ (X) is closed we may pass to the limit for → 0 on the right hand side of (17) which yields where ν is the measure associated to the form (E τ , D(E τ )) and starting point x or y respectively. Using once more that X n−2 is polar for (E, D(E)) we find that indeed lim sup By the continuity of the maps Π i : C R+ (X × X) → C R+ (X) the measures µ i ρ = Π i * (Σ T i ρ * µ) also converge to µ i = Π i * µ for ρ → 0, and since the corresponding sequence of generating Dirichlet forms (E τ ρ , D(E τ ρ )) converges in the sense of Mosco to (Eτ, D(Eτ )) the limiting measures µ i (i = 1, 2) must be the unique measures on C R+ (X) generated by (Eτ, D(E τ )) and the starting point x and y respectively.
In the case that X is non compact we have to localize the previous arguments: Fix some point 0 ∈ X and let B R (0) ⊂ X be the open metric ball around 0. Then, by the same reasoning as before we establish (16) for all functions f ∈ A ρ R : By the same arguments as above this implies where ν is the probability measure on the path space C R+ (X) induced from the Dirchlet form (E τ , D(E τ )) and Σ ρ,R is the endomorphism on C R+ (X) obtained from stopping a path when it leaves (X ∩ B R (0)) \ C ρ . analogous Dirichlet form on X ∩ B R (0)) \ C ρ ).
Using the polarity of X n−2 we may pass to the limit for ρ → 0 first, which gives In a final step we would like to pass to the limit for R → ∞. From the lower curvature bound on X it follows that and which can be transferred to the analogous statement for ν by the same argument as in (18). Hence we may send R → ∞ on both sides of (19) which concludes the proof of the proposition.
Proof. Due to elliptic regularity theory one finds u ∈ C 0 (X) ∩ C ∞ (X n−1 ) and that u satisfies the gluing condition (+). Hence we find that A k (u) → 0 locally uniformly on X \ X n−2 , where A k is approximate Laplacian operator (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-lim k →∞ (Ξ k ,(x,y) · ) * P for a suitable subsequence k and ii) ∂C ρ intersects X n−1 transversally and iii) ∂C ρ ∩ X \ X n−1 is smooth.
Proposition 2. Let (X, d) be an n-dimensional Riemannian polyhedron with with lower curvature bound −κ < 0, and for arbitrary x, y ∈ X \ X n−2 let the measure P ∞ x,y = w-lim l →∞ (Ξ l ,(x,y) · ) * P on C R+ (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.
Proof. Using proposition 3 we may proceed as in the proof of theorem 2 if we restrict of the discussion onto the set of paths stopped at time T ρ for ρ > 0. In analogy to the proof of lemma 6 the final step is to send ρ → 0 which yields the claim.
Collecting the results we may conclude that Cranston's gradient estimate holds also in the case of Riemannian polyhedra.
Theorem 4. Let (X, d) be an n-dimensional Riemannian polyhedron with lower curvature bound −κ < 0 then any nonnegative bounded function u ∈ D(E) which is weakly harmonic on (X, d) satisfies the gradient estimate (14).
Proof. This follows from lemma 6 with optional sampling, using proposition 2 and the continuity of weakly harmonic functions.
Just as in the smooth situation we can easily transfer the arguments to the coupling by parallel transport method ( * * P) which yields a corresponding gradient estimate for the heat semigroup on (X, d).