Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds
GGeodesic random walks, diffusion processes and Brownian motionon Finsler manifolds.
Tianyu Ma, Vladimir S. Matveev and Ilya Pavlyukevich
Abstract
We show that geodesic random walks on a complete Finsler manifold of bounded geometryconverge to a diffusion process which is, up to a drift, the Brownian motion corresponding toa Riemannian metric.
MSC 2000:
Key words:
Geodesic random walks, weak convergence, Finsler manifold, diffusion process,Riemannian Brownian motion, averaged metric, bounded geometry.
Contents { ξ N } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Convergence of geodesic random walks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Many processes in physics and natural sciences can be described with the help of randomwalks and their limit processes, the so-called diffusion processes. A possible philosophical ex-planation of this experimentally observed phenomenon is that the limit of random walks reflectsthe microscopic nature of the situation: Even fully deterministic microscopic systems can giverise to erratic seemingly random motions, practically indistinguishable from those produced by astochastic process. 1 a r X i v : . [ m a t h . DG ] F e b et us recall one of the first constructions of a random walk which is due to K. Pearson in1905 [33]. A more physically motivated approach is in the paper [13] of A. Einstein from thesame year.We start from a point p ∈ R , choose a random direction at the tangent space, go for distance1 along the straight line starting at this direction, and then repeat the procedure iteratively. Weobtain a stochastic process whose trajectories are piecewise-linear curves, see Fig. 1. - - - - - - Figure 1: 50 steps of a Pearson randomwalk.It is natural to renormalise this process as follows:we assume that the steps have length not 1 but 1 / √ N and we take N steps in one unit of time. If the proce-dure of choosing the random direction is invariant withrespect to the isometry group of the flat R which wasthe case in [13, 33], then by the Functional Central LimitTheorem the limit of this sequence as N → ∞ exists andis the (flat) Brownian motion , see [8, Chapter 2].We see that in order to define such a random walk,one needs two ingredients: the rule of choosing a ran-dom direction at a current position p (i.e., a probabilitydistribution ν p on the space of tangent vectors at thepoint p ) and an analogue of the notion of a straight line,which describes the motion of a small particle with noexternal forces acting upon it.In many systems in physics and natural sciences,small particles with no external forces acting upon themmove along geodesics of a Finsler metric. We give neces-sary definitions in § p of a Finslermanifold ( M, F ) such that every tangent space T p M is equipped with a probability measure ν p ,choose a random vector v in the tangent space, go the distance F ( v ) / √ N along the geodesicstarting at p with the initial velocity v and then repeat the procedure (if ν p is not centered werescale it as in § F , a volumeform µ , and an extra data u ∈ H ( M ) on M . It is easy to see that for a generic Finsler metric,solutions of this stochastic differential equation do not correspond to a limit process of a sequenceof geodesic random walks.This general approach, in which one starts with an elliptic differential operator (or a Dirichletform) in order to construct a diffusion process, is a very popular and powerful approach todiffusion processes on metric spaces. It allows in particular to treat the case of non-smoothbackground metric structures, see e.g. [18, 27, 42]. This approach does not ensure that theresulting stochastic process is the limit process of a sequence of random walks. If the backgroundis almost Riemannian (say, Alexandrov with bounded curvature, as in [18] and [27]), the bestone can do is to relate random walks on the Riemannian spaces approximating our metric spaceto the diffusion process on our metric space. These results cannot be applied in the Finsleriansituation, since Finsler metrics cannot be approximated by Riemannian metrics. Our results willpossibly allow to extend this group of methods to a Finslerian situation and we plan to do thisin our future works.Let us now discuss the corresponding results of the book [4], where many different approachesof constructing different non-equivalent diffusion processes (on the manifold or on the tangentbundle to the manifold) by a Finsler metric are suggested. One of these approaches (see [4, § A2])is seemingly close to ours, and considers the limit processes of Finsler geodesics random walks(in their case, the distribution ν p is quite special and is canonically constructed by the Finslermetric). Unfortunately no rigorous proof of convergence is given: it is merely claimed that thelimit process exists and is unique, and referred to [35, 36] for methods and technical details.The references [35, 36] are mostly survey papers about geodesic random walks on Riemannianmanifolds. The methods discussed there assume and rely on the special form of the probabilitymeasure ν p on tangent spaces. Moreover, it is assumed that the Riemannian manifold is stochas-tically complete . The property of stochastic completeness is a nontrivial property, and examplesshow that not all complete manifolds are stochastically complete. In the Riemannian case, thereis a number of criteria of stochastic completeness, see e.g. [19, 46]. In particular, if the Riccicurvature of a complete Riemannian manifold is bounded from below, the manifold is stochas-tically complete. In the Finslerian situation, we did not find any relevant works on stochasticcompleteness and the claim of [4, § A2] that the methods of [35, 36] can easily be applied in theFinslerian situation looks overoptimistic.Note that as a by-product, we have proved that every complete Finsler manifold of boundedgeometry (see Definition 2.1) is stochastically complete; that is, the limit process of Finslergeodesic random walks is stochastically complete in the sense of [21, § ν p , is in [22].Many arguments in [22] are based on the following property which holds in the Riemannian butnot in the Finslerian case: Consider an arc-length parametrized geodesic segment γ : [0 , ε ] → M ofa (smooth) Riemannian metric. Take a vector v ∈ T γ (0) M of length one and its parallel transport v ε ∈ T γ (0) M along the geodesic segment. Next, consider the arc-length parametrised geodesicgeodesics γ v and γ v ε which start from γ (0) and γ ( ε ) with the initial vectors v and v ε , respectively.Then the distance between γ v ( t ) and γ v ε ( t ), behaves, for ε → t →
0, as ε (1 + Ct ). In3he Euclidean case, the distance does not depend on t at all and is equal to ε . In the Finsleriansituation, this property does not hold for a generic metric and a straightforward generalisationof [22] is not possible.In this paper we prove that under the assumptions natural from the viewpoint of Finsler ge-ometry (everything is smooth, the manifold is complete and has bounded geometry), the sequenceof geodesic random walks converges to a unique diffusion process, see Theorem 2.1. Moreover,we show that the generator of this diffusion process is an elliptic operator, and give an integralformula for its coefficients.As explained above, the generator of the limit diffusion process is a non-degenerate ellipticoperator. If the probability measure ν p on each T p M is constructed by F | T p M (we give examplesin § Riemannian Brownian motion . This result of us explains why it is hardor even impossible to experimentally distinguish a diffusion process coming from a Riemannianmetric from that of coming from a Finsler metric. See § § Acknowledgements.
We thank M. von Renesse for useful discussions. T.M. and V.M. thank the DFG for thefinancial support (Einzelprojekt MA 2565/6).
First we recall the basic definitions in Finsler geometry. Let M := M m be a m -dimensionalmanifold, m ≥
1. Suppose that ( x , . . . , x m ) is a local coordinate at some p ∈ M . Then y i = ∂x i induces a local coordinate ( x , . . . , x m , y , . . . , y m ) on T M . For simplicity, for a function H : T M → R we use the notations H x i = ∂ x i H and H y i = ∂ y i H .A smooth Finsler manifold ( M, F ) is a smooth manifold M together with a continuous func-tion F : T M → R ≥ called the Finsler metric (Finsler function) satisfying the following condi-tions:
Regularity:
The function F is smooth on T M \ { } .4 ositive Homogeneity: For any ( x, y ) ∈ T x M and λ ≥
0, we have F ( x, λy ) = λF ( x, y ). Strong Convexity:
For 0 (cid:54) = ( x, y ) ∈ T x M , the fundamental tensor defined by[ g ( x,y ) ] ij := (cid:18) F (cid:19) y i y j (2.1)is strictly positive definite.The indicatrix bundle of ( M, F ) is defined by IM = { Y ∈ T M : F ( Y ) = 1 } . For any p ∈ M , the fibre I p M of IM is a convex hypersurface in T p M diffeomorphic to S m − .If ( M, g ) is a Riemannian manifold, one can naturally endow it with a Finsler metric by setting F ( Y ) := (cid:112) g ( Y, Y ), Y ∈ T M . Conversely, a Finsler function corresponds to some Riemannianmetric g if and only if its fundamental tensor g ij defined in (2.1) depends only on the x i -variables.The definitions of geodesics and exponential maps can be naturally generalised to the Fins-lerian situation. A smooth curve γ : [ a, b ] → M is a geodesic , if it is a stationary point of theenergy functional E [ γ ] := 12 (cid:90) ba F ( γ ( t ) , ˙ γ ( t )) d t. (2.2)among all piecewise smooth curves starting at γ ( a ) and ending at γ ( b ). It is known that for any p ∈ M and for any Y ∈ T p M , there exists a unique geodesic γ Y = γ Y ( t ) such that γ (0) = p and˙ γ (0) = Y . We define the exponential map at p to beexp p : T p M (cid:51) Y (cid:55)→ γ Y (1) ∈ M (2.3)for all Y ∈ T p M such that γ Y ( t ) is defined for t ∈ [0 , M, F ) is forward complete iffor any p ∈ M the exponential map exp p is defined for all Y ∈ T p M . The manifold ( M, F ) is geodesically complete , if each geodesic γ can be extended to a geodesic defined for all t ∈ ( −∞ , ∞ ).For a piecewise smooth curve γ : [ a, b ] → M , its length is defined by Length ( γ ) = (cid:90) ba F ( γ ( t ) , ˙ γ ( t )) d t. (2.4)The Finsler function F defines the following asymmetric and symmetrized distances on M : d a ( p, q ) := inf (cid:110) Length ( γ ) : γ is a piecewise smooth curve from p to q (cid:111) ,d ( p, q ) := max { d a ( p, q ) , d a ( q, p ) } (2.5)By the Hopf–Rinow theorem for Finsler manifolds (see e.g. [5, Section 6.6]), if ( M, F ) is forwardcomplete, the metric space (
M, d ) is complete. For a forward complete (
M, F ), every closed ballof (
M, d ) is compact. The manifold M can be naturally endowed with the Borel sigma-algebrathat makes it a measure space.Like in the Riemannian case, geodesics of Finsler metrics are local distance minimizing (withrespect to d a ) curves. The formula (2.2) ensures that they are parametrised proportional to the5rc-length parameter. Note, as F is in general not reversible , i.e. F ( x, y ) (cid:54)≡ F ( x, − y ), the distancefunction d a and geodesics are not reversible either.We will assume below that the flag and T -curvatures (the definitions are in e.g. [38]) of ourFinsler manifold are uniformly bounded. The flag curvature K can be thought as a generalisationof the Riemannian sectional curvature. The definition of T -curvature (see [38, § H c : The manifold ( M, F ) is connected and forward complete. H b : The manifold ( M, F ) has bounded geometry in the following sense:
Definition 2.1.
We say a Finsler manifold (
M, F ) has bounded geometry if the followings hold:1. Uniform ellipticity: There is some constant
C > p ∈ M and any non-zero u, v ∈ T p M , we have1 C F ( v ) = 1 C g v ( v, v ) ≤ g u ( v, v ) ≤ C g v ( v, v ) = C F ( v ) . (2.6)2. The flag curvature K is bounded uniformly and absolutely by some constant λ >
0, namely (cid:107) K (cid:107) ≤ λ .3. The T -curvature is also bounded uniformly and absolutely in the following sense. For any p ∈ M , any u, v ∈ T p M with F ( v ) = 1, the T -curvature satisfies | T v ( u ) | ≤ λ { g v ( u, u ) − [ g v ( u, v )] } (2.7)Note that all objects used in the definition of “bounded geometry” are microlocal, and for anexplicitly given Finsler metric it is possible to check whether it has bounded geometry. Moreover,if M is compact, then every smooth Finsler metric on it has bounded geometry.In this paper we use the following notations.We say a function f : M → R vanishes at infinity, if ∀ ε >
0, there exists some compact set K ε ⊂ M such that (cid:107) f (cid:107) ≤ ε outside K ε .We denote the unit discs on T M by D p M = { Y ∈ T p M : F ( Y ) ≤ } . Let d be the symmetrized distance defined by (2.5). We denote the open balls by: B p ( ε ) = { q ∈ M : d ( p, q ) < ε } , ε > . In this paper B is the space of Borel measurable real valued functions on M , C is the spaceof continuous that vanish at infinity, C ∞ is the space of smooth functions, C ∞ K is the space ofsmooth functions with compact support. Furthermore, D ([0 , ∞ ) , M ) is the collection of rightcontinuous functions γ : [0 , ∞ ) → M with left limits, and C ([0 , ∞ ) , M ) is space of continuousfunctions γ : [0 , ∞ ) → M . 6 .2 Rescaled geodesic random walks. As a motivation for “rescaling”, let us consider the following example of a geodesic randomwalk: The manifold is R with the standard flat metric and ν p is defined as follows. At everypoint p , the support of ν p is two vectors { , − } ⊂ T p R = R such that the probability of − /
4, and probability of 1 is 3 /
4. That is, the particle goes the distance 1 / √ N with probability1 / / / (2 √ N ) in thepositive direction, so in N steps the mean value is √ N /
2. For N → ∞ , most trajectories “escapeto infinity”, so the limit of the sequence of such stochastic processes for N → ∞ does not exist.This phenomenon appears in all dimensions when the probability distribution ν p (on T p M )has a nonzero mean (note that almost every trajectory of the standard flat Brownian motion isnot a rectifiable curve).Because of this, we introduce below the rescaled random walk (we will formalize this definitionin § ν p by µ p , and modify the measure ν p by shifting it in T p M by thevector − µ p + µ p / √ N , i.e. ˜ ν p ( y ) := ν p ( y − µ p + µ p / √ N ). We will call this operation the rescalingof measure .Let us explain this operation. First we note that the easiest would be to shift the measure bythe vector − µ p . This will make the mean of the new measure equal to 0, and the phenomenondemonstrated in the example above does not appear. Note that some papers on random walks onFinsler metric, for example [4], assume that both Finsler metric and the measure µ p are centrally-symmetric on every T p M (the so-called reversible situation). We do not want to do it since mostexamples of Finsler metrics appearing in applications are not reversible.This rescaling of the measure was used in the Riemannian situation by E. Jørgensen in [22],and his motivation, which is also valid in our situation, was that for N = 1 the increments of therandom walk should be distributed according to ν p . Indeed, for N = 1 we have − µ p + µ p / √ N = 0.We also feel that in the case that Finsler geodesic random walk is used to describe a physicalmodel, the mean of ν p should somehow come in the definition. Of course, it may come with anyother coefficient α , i.e., ν p may be shifted by the vector − µ p + αµ p / √ N . But also in this case(even if α depends on the position) our results are applicable. Indeed, if we modify ν p by shiftingit by βµ p , then the rescaled measure will be shifted by (1 + β ) µ p / √ N . Thus, all results of ourpaper can be applied for any α , in particular for α = 0.We also expect that such rescaling of the measure is physically-relevant, since microscopicparticles can not make too long jumps because of friction and collisions; so even if the probabilityof the particle to go to the “right” is higher than the probability to go to the “left”, the particledoes not escape to infinity in short time, contrary to what is suggested by the random walkdescribed in the beginning of this section. We will assume that the Finsler manifold (
M, F ) is connected and forward complete (Hy-pothesis H c ) and has bounded geometry (Hypothesis H b ). In addition, we make the followingassumption on the family of measures { ν p } . H ν : We assume that ν = { ν p } is a smooth family of probability measures inside DM := { Y ∈ T M | F ( Y ) ≤ } in T M or on the F -indicatrices.In the first case we require that ν is a smooth m -form on DM such that for every p , therestriction ν p := ν | T p M is a form on the disc D p M := { Y ∈ T p M | F ( Y ) ≤ } inducing a7robability measure. Similarly, in the second case ν is a smooth ( m − IM such thatfor each p ∈ M the restriction ν p := ν | I p M is a probability measure.This hypothesis is very natural from the viewpoint of Finsler geometry, and covers manychoices that have their natural counterparts in the Riemannian setting; we will give a few examplesin § Theorem 2.1.
Let Hypotheses H c , H b and H ν be satisfied. Consider a family of geodesic randomwalks starting at p constructed from ( M, F ) and { ν p } p ∈ M . Then, this sequence has a uniqueweak limit ξ . The process ξ is a diffusion whose generator is a non-degenerate elliptic differentialoperator A with smooth coefficients given by Af ( p ) = d f ( µ p ) + 12 (cid:90) T p M d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ◦ γ Y − µ p ( t ) ν p (d Y ) , f ∈ C ∞ K . (2.8) Here γ Y − µ p is the geodesic with initial vector Y − µ p . In the local coordinates, it has the followingform: Af ( p ) = f k (cid:32) µ kp − (cid:90) T p M Γ kij ( p, y − µ p ) ( y i − µ ip )( y j − µ jp ) ν p (d y ) (cid:33) + 12 f ij (cid:90) T p M ( y i − µ ip )( y j − µ jp ) ν p (d y ) , (2.9) where Γ kij are the formal Christoffel symbols of the second kind given by Γ kij ( x, y ) = 12 g ks (cid:18) ∂g is ∂x j + ∂g js ∂x i − ∂g ij ∂x s (cid:19) ( x, y ) , y (cid:54) = 0 ,f k = ∂ x k f and f ij = ∂ x i x j f .Moreover, ξ is stochastically complete. Recall that the Riemannian Brownian motion is a diffusion process which is the limit ofgeodesic random walks with identically distributed steps. Here identically distributed shouldbe understood as follows: the probability measure ν p is invariant with respect to the paralleltransport along any curve and is invariant with respect to the standard action of SO ( g ) on T p M . Actually, for a generic metric invariance with respect to the parallel transport implies SO ( g )-invariance.It is known that the generator of a Riemannian Brownian is proportional to the Beltrami–Laplace operator of the metric, so its symbol is proportional with a constant coefficient to theinverse of the Riemannian metric.By Theorem 2.1, in the Finslerian case the generator A of the limit process of the geodesicrandom walk is a second order non-degenerate elliptic differential operator on M . Hence thesymbol σ ( A ) of A is dual to a Riemannian metric on M which we denote g A . Then the Beltrami–Laplace operator ∆ A of g A and A have the same symbol. Hence A − ∆ A is just a vector field on M . We call this vector field the drift of A . In the Riemannian case the drift is always zero.In particular, though Finsler metrics are much more complicated than Riemannian metrics,one almost does not see the difference on the level of diffusion processes (only first order terms of8enerators may be different). This should be the reason why Finslerian effects related to diffusionwere not observed experimentally in physical or natural science systems, even in those where thefree motion of particles corresponds to geodesics of a certain Finsler metric. See e.g. [15] wherein a highly anisotropic situation (diffusion weighted magnetic resonance imaging of brain), themeasurement returned a Finsler metric which is very close to a Riemannian metric.This observation may provide additional mathematical tools for natural science and physics.Indeed, in most cases the probability distributions ν p can be “read” from the description of themodel (in fact in many cases they are generated by the volume form of the standard flat metric).Empirical observations of diffusions may provide tools for testing mathematical models of thesystem in question or determining their parameters. In the Riemannian situation, there is essentially only one canonical (i.e., coordinate-invariant)construction of a probability measure on T p M . Indeed, coordinate-invariance of the constructionimplies that the metric is invariant under the group SO ( g ), which implies that in the orthogonalcoordinates ( y , . . . , y m ) on T p M it is given by φ (( y ) + · · · + ( y m ) ) d y ∧ · · · ∧ d y m . Thefunction φ is the same for all points p , has the property that it is nonnegative and that theintegral (cid:82) R n φ (( y ) + · · · + ( y m ) ) d y ∧ · · · ∧ d y m = 1.In the Finslerian situations there are many natural non-equivalent constructions of a measureon T p M . Let us recall the following three. Measure coming from the Lebesgue measure:
For any p ∈ M , let ω (cid:48) p = d y ∧ · · · ∧ d y m bea Lebesgue measure on T p M . It is known that it is unique up to a positive coefficient. Werestrict it to the ball D p M (that is, the measure of an open set U ⊂ T p M it the Lebesgue-measure of the intersection D p M ∩ U ), and normalize it such that it becomes a probabilitymeasure. Measure coming from the fundamental tensor:
For any p ∈ M , the fundamental tensor g ij defines a Riemannian metric on the compact manifold I p M . Normalizing the volumeon I p M induced by g ij we obtain probability measure ν p := vol g vol g ( I p M ) . (2.10)This probability measure is close to the one used in [4, § A2].
Measure coming from the Hilbert form:
Denote P + ( M ) the positive projectivized tangentbundle. The Hilbert 1-form ˆ ω = F y i d x i defined on T M \ { } is actually a pull back ofsome 1-form ω on P + ( M ) by the standard projection. It is well known that ω ∧ (d ω ) m − defines a volume form on P + ( M ) (cid:39) IM . Let i p : I p M → IM be the standard inclusionand π : IM → M be the canonical projection. It is known (see e.g. [7]) that there exists a( m − α F on IM and a volume form ω F on M such that α F | I p M is a unique volumeform on I p M for each p ∈ M withvol α F ( I p M ) = 1 ,α F ∧ π ∗ ω F = ω ∧ (d ω ) m − . (2.11)Hence we can take ν p := vol α F on I p M . 9ach of these measures satisfies the Hypothesis H ν , and is coordinate-independently con-structed from F . In the case the Finsler metric is reversible, the dual of the symbol of thegenerator corresponding to the first measure gives the Binet–Legendre metric (see e.g. [11, 31]).The second choice of the measure gives the averaged metric used in [29, 30] (a small modificationof the construction leads to the metric from [43]), and the third choice of measure generates the
Finsler Laplacian from [7]. Note that the Binet–Legendre metric, the averaged metric, and theFinsler Laplacian from [7] appear to be effective tools for solving different problems in Finslergeometry; we expect that other natural choices of the measure ν p may also be useful in Finslergeometry. Let ( S , g ) be the unit sphere endowed with the standard Riemannian metric. Katok metric isconstructed as follows. Let X be the vector field of rotation around the axis connecting the northand south poles of the sphere such that g ( X, X ) <
1. In the the following spherical coordinateon S , ( ψ, θ ) (cid:55)→ (cos( ψ ) cos( θ ) , sin( ψ ) cos( θ ) , sin( θ )) . (2.12)We have X = r∂ ψ , (cid:107) r (cid:107) < . Now for any p ∈ M , the indicatrix I p M of the constructed Finsler function F is obtained byshifting the unit sphere S p S of g (which is the indicatrix with respect to g ) by X . That is, I p M := { v + X p : v ∈ T p S , g ( v, v ) = 1 } This yields a well-defined Finsler metric as g ( X, X ) <
1. This family of metrics depending onthe parameter r was constructed by A. Katok in [24]. It is widely used in Finsler geometry andin the theory of dynamical system as source of examples and counterexamples. It has constantflag curvature by [6, 16, 39], and by [9], any metric of constant flag curvature on the 2-sphere hasgeodesic flow conjugate to that of a Katok metric.As the measure ν p we consider the Lebesgue measure as described in § ξ .By Theorem 2.1, the diffusion process ξ generated by { ν p } p ∈ M has generator A such that Af ( p ) =d f ( X )( p ) + 12 (cid:26) f ij (cid:90) D p M ( Y i − X i )( Y j − X j ) ν p (d Y ) − f k (cid:90) D p M Γ kij ( p, Y − X ) ( Y i − X i )( Y j − X j ) ν p (d Y ) (cid:27) (2.13)Here Γ kij are the formal Christoffel symbols of the second kind of ( M, F ) and f ∈ C ∞ .As ν p is induced by a Lebesgue measure on T p M (cid:39) R m , we also denote this Lebesgue measureby ν p for simplicity. For any p ∈ M , the setˆ D P M := { Y ∈ T p M | Y + X ( p ) ∈ D p M }
10s just the closed unit ball on T p S with respect to g . Since ν p is translation invariant, Equation(2.13) becomes Af ( p ) =d f ( X )( p ) + 12 (cid:26) f ij (cid:90) ˆ D p M Y i Y j ν p (d Y ) − f k (cid:90) ˆ D p M Γ kij ( p, Y ) Y i Y j ν p (d Y ) (cid:27) (2.14)Note the integrand in the equation above is second order homogeneous in Y . By Fubini theorem,for any p ∈ M there is a finite measure η p on S p S such that for any integrable second orderhomogeneous function h on T p M , we have (cid:90) ˆ D p M h ( Y ) ν p (d Y ) = (cid:90) S p S h ( Y ) η p (d Y ) (2.15)Because ν p is invariant under any orthogonal transformation on T p S with respect to g , it isclear η p is a multiple of the canonical angular measure m p on S p S with respect to g . A straightforward computation shows η p = 14 π m p . Hence from (2.14), we get Af ( p ) = d f ( X )( p ) + 18 π (cid:26) f ij (cid:90) S p S Y i Y j m p (d Y ) − f k (cid:90) S p S Γ kij ( p, Y ) Y i Y j m p (d Y ) (cid:27) (2.16)Let ∆ be the Beltrami–Laplace operator of g , and let ˆΓ kij be the Christoffel symbols of g . Astraightforward computation yields18 ∆ f ( p ) = 18 π (cid:26) f ij (cid:90) S p S Y i Y j m p (d Y ) − f k (cid:90) S p S ˆΓ kij ( p ) Y i Y j m p (d Y ) (cid:27) . (2.17)This implies 18 ∆ and A have the same symbol. To compute the drift of A , we assume withoutloss of generality that 0 ≤ r <
1. First we have (cid:18) A −
18 ∆ (cid:19) f ( p ) = d f ( X )( p ) + 18 π f k (cid:90) S p S ˆΓ kij ( p ) Y i Y j m p (d Y ) − π f k (cid:90) S p S Γ kij ( p, Y ) Y i Y j m p (d Y ) . (2.18)Let Φ t be the flow generated by X , we know from [16, Theorem 1] that if γ ( t ) is a geodesic of( S , g ) with (cid:112) g ( ˙ γ, ˙ γ ) = c , then ˆ γ ( t ) = Φ ct ◦ γ ( t ) is a geodesic of F with initial vector ˆ γ (cid:48) (0) =˙ γ (0) + cX ( γ (0)). But in the spherical coordinate given (2.12), the flow Φ t simply has the formΦ t ( ψ, θ ) = ( ψ + rt, θ ) . Then in this coordinate, we have d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ˆ γ ( t ) = d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 γ ( t ) . By the geodesic equation, this is equivalent toΓ kij ( p, ˆ γ (cid:48) (0))(ˆ γ (cid:48) (0)) i (ˆ γ (cid:48) (0)) j = ˆΓ kij ( p )( γ (cid:48) (0)) i ( γ (cid:48) (0)) j . A given by (2.13) with r = 1 / (cid:16) A −
18 ∆ (cid:17) = X + 14 r cos( θ ) sin( θ ) · r cos ( θ ) − − r cos ( θ )) · ∂ θ . (2.19)This is the drift of the generator A .On Figure 2 one clearly sees the difference in the behaviours of the Brownian motion of theinitial round metric on S and of the diffusion process corresponding to the Katok metric with r = 1 / N . Note that the same randomseed was used in both pictures. In this section, we give a short review of the tools in Finsler geometry which will be used inour proof in later sections and formalise definitions of random geodesic walks which will allow usto apply the machinery from the theory of stochastic processes.
It is well-known that stationary points of the energy functional (2.2) are solutions of theEuler-Lagrange equation which in our situation is equivalent to the following system of ODEs:d x i d t = y i , (3.1)d y k d t + Γ kij ( x, y ) y i y j = 0 . (3.2)12ere Γ kij are the formal Christoffel symbols of the 2nd kind:Γ kij ( x, y ) = 12 g ks (cid:18) ∂g is ∂x j + ∂g js ∂x i − ∂g ij ∂x s (cid:19) ( x, y ) , y (cid:54) = 0 (3.3)It is immediate from (3.3) that for any λ > y (cid:54) = 0, we haveΓ kij ( x, y ) = Γ kij ( x, λy ) . (3.4)Denote the class of real-valued k -times continuously differentiable real-valued functions on M with compact support by C kK . Suppose f ∈ C k K with compact support K f . For any Y ∈ T M ,define f Y ( t ) = f ◦ γ Y ( t ), where γ Y ( t ) is the geodesic with initial vector Y as before. Lemma 3.1.
Suppose that f ∈ C k K . There exists some constant c such that (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:19) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cF k ( Y ) , ∀ k ≤ k , (3.5) wherever it is well defined.Proof. First we show there is some constant c such that (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:19) (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cF k ( Y ). Let ˜ X be thegeodesic spray on T M , and denote π : T M → M the canonical projection. The function π ∗ f is C k , so ( L ) k ˜ X ( π ∗ f ) is continuous on T M \ { } for any k ≤ k . Hence for some constant c , we have | ( L ) k ˜ X ( π ∗ f )(ˆ p ) | ≤ c, ∀ ˆ p ∈ IK f , ∀ k ≤ k . For any p ∈ M and Y ∈ T p M , we have (cid:18) d k d t k f Y (cid:19) (0) = ( L ) k ˜ X ( π ∗ f )( p, Y ) . In addition, for Y (cid:54) = 0, let Y (cid:48) = YF ( Y ) . Using f Y ( t ) = f ◦ γ F ( Y ) Y (cid:48) ( t ) = f Y (cid:48) ( F ( Y ) t ) , we get (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:19) (0) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:48) (cid:19) (0) (cid:12)(cid:12)(cid:12)(cid:12) F k ( Y ) . Then for p ∈ K f and Y (cid:54) = 0, we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:19) (0) (cid:12)(cid:12)(cid:12)(cid:12) = ( L ) k ˜ X ( π ∗ f )( p, Y (cid:48) ) · F k ( Y ) ≤ cF k ( Y ) . For p / ∈ K f , we have f vanishes identically on some neighbourhood of p . Then the function t (cid:55)→ f Y ( t ) is constant near t = 0. For Y = 0, then function f Y ( t ) is also constant. It follows that (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:19) (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cF k ( Y ).Next, given any geodesic γ Y , let Y ( t ) be its velocity field. We have F ( Y ( t )) = F ( Y (0)) = F ( Y ) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y (cid:19) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) d k d t k f Y ( t ) (cid:19) (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cF k ( Y ( t )) = cF k ( Y ) . This completes the proof. 13he injective radius at p is defined byinj M ( p ) := inf { r > p | D p ( r ) is injective } inj M := inf { inj M ( p ) : p ∈ M } The conjugate radius is defined similarly bycon M ( p ) := inf { r > p | ( D p ( r )) is an immersion } con M := inf { con M ( p ) : p ∈ M } . Clearly we always have inj M ( p ) ≤ con M ( p ) for any p ∈ M . The conjugate radius and flagcurvature are related by the following well known result, [5, Section 9.5]. Proposition 3.2 (Morse–Schoenberg) . Suppose ( M, F ) is a Finsler manifold such that its flagcurvature (cid:107) K (cid:107) ≤ λ . Then the conjugate radius is bounded from below by con M ≥ π √ λ , hencestrictly positive. We begin this section by a brief review of the basic definitions in Markov processes used inthis paper. Roughly speaking, a stochastic process is said to be Markovian if its future statesdepend only upon the present state, regardless of its past state.
Definition 3.1.
Let (Ω , F , P ) be a probability space. An M -valued process ξ : Ω × [0 , ∞ ) → M is Markovian if for each Borel subset of B of M , for all n ≥
1, 0 ≤ s < · · · < s n < s < t we have, P ( ξ t ∈ B | ξ s , . . . , ξ s n , ξ s ) = P ( ξ t ∈ B | ξ s ) . (3.6)The transition probability function for a Markov process is defined by P ( p, s, t, B ) = P ( ξ t ∈ B | ξ s = p ) , ∀ p ∈ M, ∀ ≤ s ≤ t. We say ( ξ t ) t ≥ is time homogeneous if the following holds. P ( p, s, t, B ) = P ( p, , t − s, B ) , ∀ p ∈ M, ∀ ≤ s ≤ t. All Markov processes considered in this paper are time-homogeneous. A time homogenous Markovprocess ξ defines a semigroup T = ( T t ) t ≥ of linear operators on the measurable functions B on M by T t ( f )( p ) = E p [ f ( ξ t )] , p ∈ M, t ≥ . We say a Markov process ξ is Feller if the semigroup T is a strongly continuous semigroup ofpositive contractions on the Banach space C .In the introduction, we gave a slightly informal definition of (rescaled) geodesic random walks.We now give a formal definition.Let ( M, F ) be a geodesically complete Finsler manifold. Let { ν p } p ∈ M be a family of measuressuch that each ν p is a probability measure on T p M . Denote by µ p the mean of ν p µ p := (cid:90) T p M Y ν p (d Y ) . In our setting (Hypothesis H ν ), the probability measures are compactly supported so µ p existsand is finite. 14 efinition 3.2. Let N ≥ p ∈ M be fixed. A random process ( ζ Nk , Y Nk +1 ) k ≥ is calleda (rescaled) discrete time geodesic random walk on M with initial point p and with increments { Y Nk +1 } k ≥ compatible with the family { ν p } p ∈ M if1. the process ζ Nk is M -valued, and Y Nk +1 is T ξ Nk M -valued,2. ζ N = p ,3. for each k ≥
0, Law( Y Nk +1 ) = ν Nζ Nk where ν Np ( B ) = (cid:90) T p M I B (cid:16) Y − µ p √ N + µ p N (cid:17) ν p (d Y ) (3.7)for any measurable B ⊆ T p M ,4. ζ Nk +1 = exp ζ Nk ( Y Nk +1 ), k ≥ ζ N are defined by the family of measures { ν p } p ∈ M and the geometryof the exponential mapping exp p . The random walks are time-homogeneous since the family { ν p } p ∈ M does not depend on k .In the classical (Euclidean) setting, random walks are processes with independent increments.In our setting the independence is understood in the conditional sense, i.e. the increments Y Nk +1 depend only on the current position ζ Nk and not on the previous positions and increments. Moreprecisely, we introduce the natural filtration F Nk := σ { ( ζ N , Y N ) , . . . , ( ζ Nk − , Y Nk ) } , k ≥ , (3.8)and say that the increments of ζ N are independent if for each f ∈ C b ( ⊕ k +1 i =0 M, R ) E (cid:104) f ( ζ N , . . . , ζ Nk , ζ Nk +1 ) (cid:12)(cid:12)(cid:12) F Nk (cid:105) = (cid:90) T ζNk M f ( ζ N , . . . , ζ Nk , exp ζ Nk ( Y )) ν Nζ Nk (d Y ) . (3.9)It is clear that ζ N is a homogeneous discrete time M -valued Markov chain with the one-steptransition operator P N f ( p ) = E p f ( ζ N ) = (cid:90) T p M f (cid:16) exp p (cid:16) Y − µ p √ N + µ p N (cid:17)(cid:17) ν p (d Y ) , f ∈ C b ( M, R ) . (3.10)Since we work in a continuous time setting it is convenient to transform the discrete timeMarkov chain ζ N into a continuous time Markov process. This can be done by a standardsubordination procedure.Let Q = ( Q t ) t ≥ be a standard Poisson process independent of { ζ N } . Define a pseudo-Poisson process ξ Nt = ζ NQ Nt , t ≥ . Note that the sample paths of ξ N belong to D ([0 , ∞ ) , M ). Hence, the Markov processes ξ N induce probability distributions P N on the path space D ([0 , ∞ ) , M ). It is easy to see that thetransition semigroup T N = ( T Nt ) t ≥ of ξ Nt has the form T Nt ( f )( p ) = E p [ f ( ξ Nt )] = e − Nt ∞ (cid:88) k =0 ( N t ) k k ! ( P N ) k ( f )( p ) , f ∈ B . (3.11)15inally we introduce a of family continuous M -valued processes defined byˆ ξ Nt = exp ζ Nk (cid:16) N (cid:16) t − kN (cid:17) Y Nk +1 (cid:17) , t ∈ (cid:104) kN , k + 1 N (cid:105) , k ≥ . Since the manifold is geodesically complete, the processes ζ N , ξ N and ˆ ξ N are well-defined foreach N ≥
1. By construction the processes ˆ ξ N have piecewise smooth sample paths consisting ofgeodesic segments, and induce probability distributions ˆ P N on the path space C ([0 , ∞ ) , M ) ofcontinuous M -valued functions. These are the geodesic random walks introduced and discussedin the Introduction, see Fig. 1 there, and in § ξ N arenot Markovian, the convergence of the continuous time processes ( ζ N [ Nt ] ) t ≥ , ( ξ Nt ) t ≥ and ( ˆ ξ Nt ) t ≥ is equivalent, see, e.g. [23, Theorem 17.28]. In the next sections we will mainly work with theMarkov processes ξ N . In this section we prove the convergence of the geodesic random walks { ξ N } . We will assumethat the Finsler manifold ( M, F ) is forward complete and connected (Hypothesis H c ) and hasbounded geometry (Hypothesis H b ) and that the measures { ν p } p ∈ M satisfy the condition H ν from § In this section we show the N -scaled geodesic random walks on a complete Finsler manifold( M, F ) with bounded geometry are Feller.
Lemma 4.1.
Let H ν hold true and k ≥ . For any C k -smooth function f : T M → R the mapping p → (cid:90) T p M f ( Y ) ν p (d Y ) (4.1) is also C k -smooth. This lemma is obvious since each ν p is only supported on D p M . Indeed, integral over a com-pact set of a function smoothly depending on parameters smoothly depends on the parameters.Now we are ready to show the semigroups { T N } are Feller and give the formula of thegenerators. Proposition 4.2.
Suppose ( M, F ) is complete and uniform elliptic. In addition, assume themeasures { ν p } satisfy the hypothesis H ν . Then for each N ≥ , the family of operators T N =( T Nt ) t ≥ is a conservative Feller semigroup with the generator A N f = N (cid:16) P N f − f (cid:17) , f ∈ C . (4.2) Proof.
Let N ≥ T N is a strongly continuous semigroup ofa pseudo-Poisson process, its generator has the form (4.2) by Theorem 19.2 from [23]. It isconservative due to assumption H c . Let us show that T Nt maps C into itself for t ≥
0. Since wehave (cid:107) P N f (cid:107) ≤ (cid:107) f (cid:107) , (4.3)16he series in (3.11) converges uniformly. It suffices to show P N maps C into itself.By Lemma 4.1 the mean value µ p is a C ∞ vector field. Since the exponential map for Finslermanifold is at least C , then P N maps continuous functions into continuous functions.For any ε >
0, choose some compact K ⊆ M such that | f ( x ) | < ε x / ∈ K . Fix any p ∈ K ,and define the closed forward balls at p for R ≥ B + p ( R ) := { q ∈ M : d a ( p , q ) ≤ R } . Because K is compact, there exists some R > K ⊂ B + p ( R ) for all R ≥ R . Bythe Hopf–Rinow theorem (see Theorem 6.6.1 of [5]), the forward closed balls B + p ( R ) are alsocompact.For any 0 (cid:54) = Y ∈ T p M and p ∈ M , the uniform ellipticity condition in Definition 2.1 gives F ( − Y ) = g − Y ( Y, Y ) ≤ C g Y ( Y, Y ) = C F ( Y ) . (4.4)It follows that d a ( q, p ) ≤ Cd a ( p , q ) ≤ CR , ∀ q ∈ K ; (4.5) d a ( p, p ) ≥ C d a ( p , p ) ≥ RC , ∀ p ∈ ( B + p ( R )) c , ∀ R ≥ . (4.6)Let R := C ( C + 2 + CR ), then ∀ p ∈ ( B + p ( R )) c and ∀ q ∈ K , we have d a ( p, q ) ≥ d a ( p, p ) − d a ( q, p ) ≥ R C − CR > C + 1 (4.7)On the other hand, for p ∈ M and Y ∈ D p M , we have d a ( p, e Np ( Y )) ≤ F ( 1 √ N ( Y − (1 − / √ N ) µ p )) ≤ F ( Y ) + F ( − µ p ) ≤ C + 1 . (4.8)Hence, ∀ Y ∈ D p M and p ∈ ( B + p ( R )) c , we have e Np ( Y ) / ∈ K . It follows that ∀ p ∈ ( B + p ( R )) c : | P N f ( p ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T p M f ◦ e Np ( Y ) ν p (d Y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . (4.9)That is to say (cid:107) P N f (cid:107) ≤ ε B + P ( R ). We conclude that P N f ∈ C . Thiscompletes the proof. In this section, we prove the generators A N converge on the space C ∞ K to some second orderelliptic operator with smooth coefficients.Denote f Y − µ p ( t ) = f ◦ γ Y − µ p ( t ) (4.10) Proposition 4.3.
Let A be the differential operator defined by Af ( p ) := d f ( µ p ) + 12 (cid:90) T p M d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 f Y − µ p ( t ) ν p (d Y ) , f ∈ C . (4.11) Then A is a second order positive definite elliptic operator of smooth coefficients and for each f ∈ C ∞ K lim N →∞ (cid:107) A N f − Af (cid:107) = 0 . (4.12)17 roof. The proof follows the steps from [22] in the Riemannian case. By computing the Taylorexpansion of A N f , we show the convergence of the first and second order terms and vanishing ofother higher order terms as N → ∞ .Take any f ∈ C ∞ K . We have A N ( f )( p ) = N (cid:16) P N ( f )( p ) − f ( p ) (cid:17) = N (cid:90) T p M (cid:104) f ◦ γ Y − (1 − / √ N ) µ p (cid:16) √ N (cid:17) − f ( p ) (cid:105) ν p (d Y ) (4.13)Then for any p ∈ M and Y ∈ D p M , the Taylor expansion of f Y − (1 − / √ N ) µ p ( t ) = f ◦ γ Y − (1 − / √ N ) µ p ( t ) (4.14)gives f Y − (1 − / √ N ) µ p (cid:18) √ N (cid:19) = f ( p ) + 1 √ N d f p ( Y − (1 − / √ N ) µ p )+ 12 N d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:16) f Y − (1 − / √ N ) µ p ( t ) (cid:17) + R N ( p, Y ) . Thus we have A N f ( p ) = N (cid:90) T p M (cid:26) √ N d f p ( Y − (1 − / √ N ) µ p )+ 12 N d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f Y − (1 − / √ N ) µ p ( t )) + R N ( p, Y ) (cid:27) ν p (d Y )= d f ( µ p ) + (cid:90) T p M
12 d d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f Y − (1 − / √ N ) µ p ( t )) ν p (d Y ) + (cid:90) T p M N R N ( p, Y ) ν p (d Y ) . (4.15)Using Lemma 3.1 and equation (4.4), for any Y ∈ D p M and p ∈ M , there is some constant c f > | R N ( p, Y ) | ≤ N √ N sup t ∈ [0 , / √ N ] (cid:12)(cid:12)(cid:12)(cid:12) d d t f Y − (1 − / √ N ) µ p ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c f N √ N F ( Y − (1 − / √ N ) µ p ) ≤ c f N √ N ( F ( Y ) + F ( − µ p )) ≤ c f N √ N ( C + 1) . (4.16)Clearly, we have from (4.16): lim N →∞ sup p ∈ M,Y ∈ D p M | N R N ( p, Y ) | = 0 . (4.17)The last term in (4.15) tends to zero, since ν p is only supported on D p M .18or the second order term, in a canonical coordinate of T M , we haved d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f Y − (1 − / √ N ) µ p ( t ))= f ij · y i (cid:18) Y − (1 − √ N ) µ p (cid:19) y j (cid:18) Y − (1 − √ N ) µ p (cid:19) − f k · Γ kij (cid:18) p, Y − (1 − √ N ) µ p (cid:19) y i (cid:18) Y − (1 − √ N ) µ p (cid:19) y j (cid:18) Y − (1 − √ N ) µ p (cid:19) . (4.18)Since the formal Christoffel symbols are bounded on each compact local coordinate, the right-hand-side of (4.18) converges to f ij · y i ( Y − µ p ) y j ( Y − µ p ) − f k · Γ kij ( p, Y − µ p ) y i ( Y − µ p ) y j ( Y − µ p ) , as N → ∞ uniformly on DK for each compact chart K ⊆ M , where DK = { Y ∈ D p M : p ∈ K } .Choose a smooth coordinate on some open U ⊆ M . The chain rule implies A has the followingform in this coordinate. Af ( p ) =d f ( µ p ) + 12 (cid:18) f ij (cid:90) T p M y i ( Y − µ p ) y j ( Y − µ p ) ν p (d Y ) − f k (cid:90) T p M Γ kij ( p, Y − µ p ) y i ( Y − µ p ) y j ( Y − µ p ) ν p (d Y ) (cid:19) . Because f has compact support, we havelim n →∞ (cid:107) A N f − Af (cid:107) = 0 , f ∈ C ∞ K . (4.19)It follows that the symbol of A is: σ ( A )( p ) = 12 (cid:90) T p M ⊗ ( Y − µ p ) ν p (d Y ) . (4.20)For each p ∈ M , the measure ν p is induced by either a smooth non-zero ( m − I p M oran m -form T p M (condition H ν ). Then σ ( A ) is strictly positive definite, and hence A is a strictlyelliptic operator.In each compact local coordinate, the functions { Γ kij } are bounded, and smooth on T M \ { } .It follows that A has smooth coefficients. This completes the proof. { ξ N } . In this section we prove the family of random walks { ξ N } is tight in D ([0 , ∞ ) , M ). Recallthat the symmetrized distance d makes ( M, d ) a complete separable metric space, as (
M, F ) isforward complete and has bounded geometry.
Proposition 4.4.
Let the Finsler manifold ( M, F ) and the family { ν p } p ∈ M satisfy Assumptions H c , H b and H ν . Then the family of random walks { ξ N } N ≥ is tight in D ([0 , ∞ ) , M ) .Proof. The statement follows from the Aldous criteria (Lemma 4.8) and the compact containmentcondition (Lemma (4.9)) that will be proven in this section.19ur goal consists of obtaining uniform estimates for the oscillation of the random walks ξ N ,see Equation (4.53). Because M is in general non-compact, the injective radius lower boundinj M can be zero. Since the non-symmetrized distance function d a ( p, · ) is smooth only withinthe injective radius, we work on the tangent bundle T M to bypass this technical problem. Toprepare the proof of Lemma 4.8 as well as Lemma 4.9, for R ≥
0, define D p ( R ) := { Y ∈ T p M : F ( Y ) ≤ R } . We make the following construction.The condition K ≤ λ implies there exists some 0 < δ c < M > δ c , see Proposition 3.2. For each p ∈ M , the exponential map exp p is a smooth immersionon D p ( δ c ) except possibly at 0. Then we can construct a geodesically complete smooth Finslerfunction F p on T p M such that F p = (exp p ) ∗ ( F ) on D p ( δ c ), while F p is the standard Minkowskimetric on T p M \ D p (1), under any standard identification T p M (cid:39) R m . To distinguish it from thedistance functions on ( M, F ), we denote the asymmetric and symmetric distance on ( T p M, F p )by d pa and d p , respectively. Note that the injective radius of F p at 0 ∈ T p M is at least δ c .Now for each p ∈ M we construct the measures { ˜ ν q } q ∈ T p M , so that on D p ( δ c / { ˜ ν q } q ∈ D p ( δ c / are the lift of { ν o } o ∈ M by the exponential map exp p . In addition, we require themeasures { ˜ ν q } q ∈ T p M satisfy the condition H ν .Then for each p ∈ M and N ≥
1, we construct an N -scaled geodesic random walk ξ N,p on theFinsler manifold ( T p M, F p ) starting at 0 ∈ T p M , using the prescribed measures { ˜ ν q } q ∈ T p M as inSection 3.2. Note as ( T p M, F p ) satisfies H b and H c , all results we proved earlier are true for therandom walks ξ N,p . Lemma 4.5.
There exists some δ > so that for each δ ∈ (0 , δ ) , there exists a family offunctions { f δp } p ∈ M such that1. Each f δp is a function on T p M with ≤ f δp ≤ such that f δp (0) = 1 and f δp ( q ) = 0 if q / ∈ D p ( δ ) .2. Denote A N,p the generator associated to ξ N,p , then there exists a constant ˜ C ( δ ) > suchthat sup N ≥ sup p ∈ M sup q ∈ T p M (cid:12)(cid:12)(cid:12) A N,p f δp ( q ) (cid:12)(cid:12)(cid:12) ≤ ˜ C ( δ ) . (4.21) Proof.
The general scheme to prove this lemma is as follows. We construct the family of functions { f δp } using the distance functions d pa (0 , · ) on ( T p M, F p ). The Hessian comparison theorem from[38, Section 15.1] applied to Finsler manifolds with bounded flag and T -curvature suggests thedistance functions have uniformly bounded Hessians. This fact applied to the Taylor expansionof f δp implies the family of functions we constructed satisfies the conditions listed in the Lemma.For δ c ∈ (0 ,
1) chosen above,
C > λ > < δ < min { δ c C +1) , π √ λ } .Fix any p ∈ M , and denote d pa (0 , · ) the distance function for ( T p M, F p ) from 0 ∈ T p M . Becausethe injective radius for F p at 0 is at least δ c , the distance function d pa (0 , · ) is smooth on the openset D p ( δ ) \ { } for each 0 < δ < δ .Fix any δ ∈ (0 , δ ). Let ψ δ : R → R be a smooth function with compact support contained in[ − δ , δ ]. Further suppose 0 ≤ ψ δ ≤ ψ δ ≡ I = [ − δ , δ ]. For each p ∈ M , the function f δp ( q ) := ψ δ ◦ d pa (0 , q ) , q ∈ T p M. (4.22)20s smooth on T p M and satisfies condition 1.To prove condition 2, for any q ∈ ( T p M, F p ), let ˜ µ q be the mean of ˜ ν q . An argument similarto Proposition 4.2 shows that for any q ∈ T p MA N,p f δp ( q ) = N (cid:104) (cid:90) T q ( T p M ) f δp ◦ γ Y − (1 − / √ N )˜ µ q (cid:16) √ N (cid:17) ˜ ν q (d Y ) − f δp ( q ) (cid:105) . (4.23)Note { ˜ ν q } q ∈ T p M satisfies H ν , so we only need to integrate over Y ∈ T q ( T p M ) with F p ( Y ) ≤ Y ∈ T q ( T p M ) h N ( Y )( t ) := d a (cid:16) , γ Y − (1 − / √ N ) µ q ( t ) (cid:17) , (4.24) h δN ( Y )( t ) := ψ δ ◦ h N ( Y )( t ) = f δp ◦ γ Y − (1 − / √ N ) µ q ( t ) , t ≥ . (4.25)By Taylor theorem, there exist functions { t N } with t N : T q ( T p M ) → (cid:16) , √ N (cid:17) (4.26)such that h δN ( Y ) (cid:16) √ N (cid:17) = f δp ( q ) + d f δp ( q ) (cid:16) Y − ˜ µ q √ N + 1 N ˜ µ q (cid:17) + 12 N d d t (cid:12)(cid:12)(cid:12)(cid:12) t = t N ( Y ) (cid:16) h δN ( Y )( t ) (cid:17) . (4.27)Using Equations (4.23) and (4.27), we get A N f δp ( q ) = N (cid:90) T q ( T p M ) (cid:104) d f δp ( q ) (cid:18) Y − ˜ µ q √ N + 1 N ˜ µ q (cid:19) + 12 N d d t (cid:12)(cid:12)(cid:12)(cid:12) t = t N ( Y ) (cid:16) h δN ( Y )( t ) (cid:17) (cid:105) ˜ ν q (d Y )= d f δp ( q )(˜ µ q ) + 12 (cid:90) T q ( T p M ) d d t (cid:12)(cid:12)(cid:12)(cid:12) t = t N ( Y ) (cid:16) h δN ( Y )( t ) (cid:17) ˜ ν q (d Y ) . (4.28)We need to show the equation above is uniformly bounded for all p ∈ M , q ∈ T p M and N ≥ < δ < δ ,sup p ∈ M sup q ∈ T p M | d f δp ( q )(˜ µ q ) | < ∞ . (4.29)Clearly if d pa (0 , q ) ≤ δ or d pa (0 , q ) ≥ δ , we haved f δp ( q )(˜ µ q ) = 0 . For any q ∈ T p M such that δ ≤ d a (0 , q ) ≤ δ , the Finlser metric F p | T q ( T p M ) and the measure ˜ ν q are the pull backs of F and { ν o } o ∈ M by exp p , respectively. Thus F p (˜ µ q ) < F p ( − ˜ µ q ) ≤ C .It follows that | ˜ µ q ( d pa (0 , · )) | ≤ C if δ ≤ d pa (0 , q ) ≤ δ . The function d pa (0 , · ) is smooth at q For q ∈ T p M with δ ≤ d a (0 , q ) ≤ δ . Hence Equation (4.29)holds by the chain rule. 21ow it suffices to prove the integrand in (4.28) is uniformly bounded for all Y ∈ T q ( T p M ), q ∈ T p M , p ∈ M and N ≥
1. By the construction of ψ δ , for the case d pa (cid:16) , γ Y − (1 − / √ N )˜ µ q ( t N ( Y )) (cid:17) ≥ δ d pa (cid:16) , γ Y − (1 − / √ N )˜ µ q ( t N ( Y )) (cid:17) ≤ δ , (4.30)we have d d t (cid:12)(cid:12)(cid:12)(cid:12) t = t N ( Y ) h δN ( Y )( t ) = 0 . (4.31)For the case δ ≤ d pa (cid:0) , γ Y − (1 − / √ N )˜ µ q ( t N ( Y )) (cid:1) ≤ δ , (4.32)the function h N ( Y )( t ) is smooth on some interval containing t = t N ( Y ), because d pa (0 , · ) is smoothon D p ( δ ) \ { } . Since ψ δ is in C ∞ K , it is sufficient to show the first and second derivatives h N ( Y )( t )with respect to t are uniformly bounded.To simplify the notations we denote by ∇ ρ := ∇ d pa (0 , · ) the Finsler gradient , see, e.g. [38,Equation (3.14) in § § g := ˜ g ∇ ρ , r N ( Y ) := d pa (cid:16) , γ Y − (1 − / √ N )˜ µ q ( t N ) (cid:17) , ˙ γ N := dd t (cid:12)(cid:12)(cid:12) t = t N ( Y ) γ Y − (1 − / √ N )˜ µ q ( t ) , Y ⊥ := ˙ γ N − ˆ g ( ˙ γ N , ∇ ρ ) ∇ ρ. Here ˜ g is the fundamental tensor of F p .Because exp p is an isometric immersion on D p ( δ ), on ( T p M, F p ), we also have the uniformelliptic conditions 1 C ˜ g v ( v, v ) ≤ ˜ g u ( v, v ) ≤ C ˜ g v ( v, v ) , (4.33)for any 0 (cid:54) = u, v ∈ T q ( T p M ) with q ∈ D p ( δ ). It follows that for all Y ∈ T q ( T p M ) with F p ( Y ) ≤ q ∈ D p ( δ ) we have F p ( ˙ γ N ) = F p (cid:16) Y − (1 − / √ N )˜ µ q (cid:17) ≤ C + 1 ,F p ( − ˙ γ N ) ≤ C + 1 . This implies that for all Y ∈ T q ( T p M ) with F p ( Y ) ≤ (cid:12)(cid:12)(cid:12) dd t (cid:12)(cid:12)(cid:12) t = t N ( Y ) h N ( Y )( t ) (cid:12)(cid:12)(cid:12) ≤ C + 1 , if (4.32) holds.The second derivative of h N ( Y )( t ) can be estimated by the Hessian comparison theorem(see Section 15.1 of [38]), using the bounded curvature conditions in Definition 2.1. Note that( D p ( δ ) , F p ) also has flag curvature and T -curvature bounded by | K | ≤ λ and | T | ≤ λ , becauseexp p restricted to ( D p ( δ ) , F p ) is an isometric immersion. Since the injective radius of F p at 0 isat least δ c > δ , the Hessian comparison theorem implies: (cid:16) √ λ · cot( √ λ · r ( Y )) − λ (cid:17) ˆ g ( Y ⊥ , Y ⊥ ) ≤ d d t (cid:12)(cid:12)(cid:12)(cid:12) t = t N ( Y ) h N ( Y ( t )) , (4.34)d d t (cid:12)(cid:12)(cid:12) t = t N ( Y ) h N ( Y ( t )) ≤ (cid:16) √ λ · coth( √ λ · r ( Y )) + λ (cid:17) ˆ g ( Y ⊥ , Y ⊥ ) . (4.35)22sing the fact ˆ g ( ∇ ρ, ∇ ρ ) = F p ( ∇ ρ ) = 1 on D p ( δ ) \ { } , we getˆ g ( Y ⊥ , Y ⊥ ) = (cid:12)(cid:12) ˆ g ( ˙ γ N , ˙ γ N ) − ˆ g ( ˙ γ N , ∇ ρ ) (cid:12)(cid:12) . (4.36)If x ∈ D p ( δ ), for any tangent vectors Y , Y ∈ T x ( T p M ) with Y (cid:54) = 0, the fundamental inequalityin Finsler geometry (see 1.2.16 of [5]) and the inequality F p ( Y ) ≤ CF p ( − Y ) give | ˜ g Y ( Y , Y ) | ≤ CF p ( Y ) F p ( Y ) . (4.37)Substituting this into (4.36) and using uniform ellipticity, for Y such that F p ( Y ) ≤ g ( Y ⊥ , Y ⊥ ) ≤ C ˜ g ˙ γ N ( ˙ γ N , ˙ γ N ) + C F p ( ˙ γ N ) ≤ C F p (cid:16) Y − (cid:16) − √ N (cid:17) ˜ µ q (cid:17) ≤ C ( C + 1) . Since δ ≤ r N ( Y ) ≤ δ and δ < π √ λ , we have for all N ≥ ≤ cot( √ λ · r N ( Y )) ≤ coth( √ λ · r N ( Y )) . (4.38)Then for all N ≥ Y with F p ( Y ) ≤
1, we have the estimate: (cid:12)(cid:12)(cid:12)(cid:12) d d t (cid:12)(cid:12)(cid:12) t = t N ( Y ) h N ( Y ( t )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) √ λ · coth( δ √ λ λ (cid:17) ˆ g ( Y ⊥ , Y ⊥ ) , (4.39) ≤ C ( C + 1) (cid:16) √ λ · coth( δ √ λ λ (cid:17) . (4.40)This shows the second derivative of h N ( Y )( t ) evaluated at t = t N ( Y ) is also uniformly andabsolutely bounded, if (4.32) holds true. Then there exists some ˜ C ( δ ) > { f δp } will be used now to estimate the first exit time from a δ -balls ofthe geodesic random walk ξ N .For each p ∈ M , N ≥ δ >
0, define the following stopping times for the random walk ξ N on M and ξ N,p on T p M : τ N,δ := inf { t > d ( ξ Nt , p ) > δ } , (4.41) τ N,δp := inf { t > d p ( ξ N,pt , > δ } , ∈ T p M. (4.42)Now we compare the exit time probabilities of the δ -balls for ξ Nt and ξ N,pt for sufficiently large N . Lemma 4.6.
For any p ∈ M , N ≥ and δ such that < δ < δ and C +1) √ N < δ c , we have P p ( τ N,δ ≤ t ) ≤ P ( τ N,δp ≤ t ) , ∀ t ≥ . (4.43)23 roof. The geodesic random walks ξ N,p and ξ N are constructed by randomizing the time of thediscrete Markov processes ζ N,p and ζ N using a Poisson process, respectively. Hence it suffices toshow for each pair ( N, δ ) satisfies the condition in the lemma, the following holds P (cid:18) max j ≤ k d p (cid:16) , ζ N,pj (cid:17) ≤ δ (cid:19) ≤ P p (cid:18) max j ≤ k d ( p, ζ Nj ) ≤ δ (cid:19) , ∀ p ∈ M, ∀ δ < δ , k ≥ . (4.44)For r >
0, define the closed δ -balls of the symmetrized distances on T p M and M respectively: B p ( r ) = { y ∈ T p M : d p (0 , y ) ≤ r } ,B p ( r ) = { q ∈ M : d ( p, q ) ≤ r } . Because δ < δ < δ c C +1) , we have B p ( δ ) ⊂ D p ( δ c / p maps ( B p ( δ ) , F p ) inside( B p ( δ ) , F ) by an isometric immersion. Now for each k ≥ B p ( δ ) and B p ( δ ), respectively. θ k ( ˆ E ) = P (cid:16) ζ N,pj ∈ B p ( δ ) , ≤ j ≤ k − , ζ N,pk ∈ ˆ E (cid:17) , ∀ ˆ E ∈ B ( B p ( δ )); θ k ( E ) = P p (cid:0) ζ Nj ∈ B p ( δ ) , ≤ j ≤ k − , ζ Nk ∈ E (cid:1) , ∀ E ∈ B ( B p ( δ )) . Let θ pk := (cid:16) exp p | B p ( δ ) (cid:17) ∗ θ k . For integers N such that C +1) √ N < δ c , we claim θ pk ≤ θ k for all k ≥ k = 0, we have θ ( ˆ E ) = ˆ E (0) and θ ( E ) = E ( p ).Since exp p (0) = p , this claim holds for k = 0. For simplicity we denote e Np := exp p (cid:16) Y − µ p √ N + 1 N µ p (cid:17) . (4.45)For any q ∈ B p ( δ ) and Y ∈ T q ( T p M ) with F p ( Y ) ≤
1, the condition C +1) √ N < δ c implies the N -scaled geodesic segment γ ( t ) = exp q ( t e Nq ( Y )) for t ∈ [0 ,
1] of F p is mapped by exp p to a geodesicsegment on ( M, F ). Also note for q ∈ B p ( δ ), the mean is preserved under exp p by(d exp p ( q )) ∗ (˜ µ q ) = µ exp p ( q ) . Hence for any q ∈ B p ( δ ) and Y ∈ T q ( T p M ) with F p ( Y ) ≤ p (cid:0) e Nq ( Y ) (cid:1) = e N exp p ( q ) (cid:0) (d exp p ( q )) ∗ ( Y ) (cid:1) . (4.46)For each N ≥
1, let P o ( x, · ) and P ( y, · ) be the one-step transition probabilities of ζ N,p and ζ N , respectively. For q ∈ B p ( δ ), set q = exp p ( q ). Since ˜ ν q is the pull-back of ν q by ( d exp p ( q )),then (4.46) implies for any Borel set ˆ E ⊂ B p ( δ ), we have P o ( q, ˆ E ) = ˜ ν q (cid:16)(cid:0) e Nq (cid:1) − ( ˆ E ) (cid:17) ≤ ν q (cid:16)(cid:0) e Nq (cid:1) − (exp p ( ˆ E )) (cid:17) = P ( q , exp p ( ˆ E )) . (4.47)The inequality in the previous formula appears because the exponential map exp p restricted to B p ( δ ) is not necessarily injective.In particular, for any Borel E ⊂ B p ( δ ), we get P ( q , E ) ≥ P o ( q, (exp p ) − ( E ) ∩ B p ( δ )) . (4.48)24y the Markov property, we have for all k ≥ θ ok ( ˆ E ) = (cid:90) B p ( δ ) P o ( y, ˆ E ) θ ok − (d y ) , ∀ ˆ E ∈ B ( B p ( δ )); θ k ( E ) = (cid:90) B p ( δ ) P ( x, E ) θ k − (d x ) , ∀ E ∈ B ( B p ( δ )) . Hence we have the following chain of inequalities: θ k ( E ) = (cid:90) B p ( δ ) P ( x, E ) θ k − (d x ) ≥ (cid:90) B p ( δ ) P ( x, E ) θ pk − (d x )= (cid:90) B p ( δ ) P (exp p ( y ) , E ) θ ok − (d y ) ≥ (cid:90) B p ( δ ) P o ( y, (exp p ) − ( E ) ∩ B p ( δ )) θ ok − (d y )= θ pk ( E ) . where the first inequality comes for the induction assumption, and the second inequality is dueto (4.48).In particular, for all k ≥
0, the inequality θ ok ( B p ( δ )) = θ pk ( B p ( δ )) ≤ θ k ( B p ( δ ))implies (4.44) holds. This completes the proof.Next, we prove the following estimate on the first exit times of ξ N from δ -balls on M . Lemma 4.7.
For each δ > there is C ( δ ) > such that for all t ≥ p ∈ M sup N ≥ P p ( τ N,δ ≤ t ) ≤ C ( δ ) t. (4.49) In particular, sup p ∈ M sup N ≥ E p e − τ N,δ < . (4.50) Proof.
We adapt ideas from [26] by Kunita. The proof is divided into two steps. First we considerthe exit times for the lifted random walks ξ N,p , and we showsup p ∈ M sup N ≥ P ( τ N,δp ≤ t ) ≤ ˜ C ( δ ) t, (4.51)where 0 ∈ T p M and ˜ C ( δ ) > δ . For any δ ∈ (0 , δ ) and p ∈ M , let { f δp } bethe family of functions constructed in Lemma 4.5. Using 0 ≤ f δp ≤ f δp (0) = 1, we have P ( τ N,δp ≤ t ) = 1 − P ( τ N,δp > t ) ≤ − E (cid:104) I ( τ N,δp > t ) f δp ( ξ N,p ( τ N,δp ∧ t )) (cid:105) , = 1 − E (cid:104)(cid:16) − I ( τ N,δp ≤ t ) (cid:17) f δp (cid:16) ξ N,p ( τ N,δp ∧ t ) (cid:17)(cid:105) = f δp (0) − E f δp (cid:16) ξ N,p ( τ N,δp ∧ t ) (cid:17) + E (cid:104) I ( τ N,δp ≤ t ) f δp (cid:16) ξ N,p ( τ N,δp ∧ t ) (cid:17)(cid:105) = f δp (0) − E f δp (cid:16) ξ N,p ( τ N,δp ∧ t ) (cid:17) + E (cid:104) I ( τ N,δp ≤ t ) f δp (cid:16) ξ N,p ( τ N,δp ) (cid:17)(cid:105) . Here the notation a ∧ b := min { a, b } is standard in the theory of stochastic processes. Takinginto account that f δp (cid:16) ξ N,p (cid:16) τ N,δp (cid:17)(cid:17) = 0 and applying the Dynkin formula to above, we obtain P ( τ N,δp ≤ t ) ≤ − E (cid:90) τ N,δp ∧ t A N,p f δp ( ξ N,ps ) d s ≤ sup N ≥ (cid:107) A N,p f δp (cid:107) · t = ˜ C ( δ ) t, ∀ p ∈ M. Let N be the smallest positive integer such that 2( C + 1) √ N < δ c N ≤ N , we have alwayshave P p ( τ N,δ ≤ t ) ≤ P ( Q ( N t ) > ≤ N t ≤ N t. (4.52)This together with Lemma 4.6 proves the inequality (4.49) by setting C ( δ ) = max { ˜ C ( δ ) , N } .Furthermore, for t ∗ = 12 C ( δ ) > E p e − τ N,δ = E p I ( τ N,δ ≤ t ∗ )e − τ N,δ + E p I ( τ N,δ > t ∗ )e − τ N,δ ≤ P p ( τ N,δ ≤ t ∗ ) + e − t ∗ (cid:16) − P p ( τ N,δ ≤ t ∗ ) (cid:17) = e − t ∗ + (1 − e − t ∗ ) P p ( τ N,δ ≤ t ∗ ) . Note that due to (4.49), we have P p ( τ N,δ ≤ t ∗ ) ≤ C ( δ ) t ∗ ≤
12 . Hence we obtain E p e − τ N,δ ≤ e − t ∗ + 1 − e − t ∗ − t ∗ < . This proves the second inequality of the lemma.Now we are ready to show the Aldous criteria hold in our situation.
Lemma 4.8 (Aldous criteria) . For any initial point p ∈ M , any T > , δ > , and any ( F N ) -stopping times ≤ τ ≤ T , we have lim s → lim sup N →∞ sup τ sup h ∈ [0 ,s ] P p (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ (cid:1) = 0 (4.53)26 roof. Let δ, s > N ≥ p ∈ M , h ∈ [0 , s ] and a stopping time τ we have P p (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ (cid:1) = E p (cid:20) I (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ ) (cid:1)(cid:21) , = E p (cid:20) E (cid:20) I (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ ) (cid:1) |F Nτ (cid:21)(cid:21) , = E p (cid:20) P (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ |F Nτ (cid:1)(cid:21) . The strong Markov property of ξ N yields E p (cid:20) P (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ |F Nτ (cid:1)(cid:21) = E p P ξ Nτ (cid:18) d ( ξ N , ξ Nh ) > δ (cid:19) ≤ E p (cid:104) sup q ∈ M P q ( τ N,δ ≤ h ) (cid:105) . Thus by Lemma 4.7, we have P p (cid:0) d ( ξ Nτ , ξ Nτ + h ) > δ (cid:1) ≤ C ( δ ) h ≤ C ( δ ) s. (4.54)Taking supremums and letting s →
0, we obtain the limit in Equation (4.53).Next we show the family of processes { ξ N } has compact containment property as follows. Lemma 4.9 (compact containment condition) . For any ε > , T ≥ and p ∈ M there is acompact neighborhood K ε ( p ) ⊆ M of p such that inf N P p (cid:16) ξ Nt ∈ K ε ( p ) , t ∈ [0 , T ] (cid:17) ≥ − ε. (4.55) Proof.
Let us define the following sequence of exit times τ N := 0 ,τ Nk := inf (cid:110) s > τ Nk − : d (cid:16) ξ Ns , ξ Nτ Nk − (cid:17) > (cid:111) , k ≥ , (as usual, we set inf ∅ = + ∞ ).By Lemma 4.7, there exists some constant c ∈ (0 ,
1) such thatsup p ∈ M sup N E p e − τ N = sup p ∈ M sup N E p e − τ N,δ ≤ c < k ≥ ∀ N ≥
1, the strong Markov property yields E p e − τ Nk = E p (cid:2) e − τ Nk − · e τ Nk − − τ Nk (cid:3) , = E p (cid:20) e − τ Nk − · E (cid:2) e τ Nk − − τ Nk |F Nτ Nk − (cid:3)(cid:21) , = E p (cid:20) e − τ Nk − · E ξ NτNk − e − τ N (cid:21) , ≤ c · E p e − τ Nk − ≤ c k . ε > T ≥
0, define k ε := (cid:24) ln ε − T ln c (cid:25) . (4.56)Then the exponential Markov inequality gives ∀ p ∈ M , ∀ N ≥ P p (cid:0) τ Nk ε ≤ T (cid:1) = P p (cid:16) e − τ Nkε ≥ e − T (cid:17) ≤ e T E p e − τ Nkε ≤ e T c k ε ≤ ε. (4.57)By construction and the triangle inequality we have that for each k ≥
1, each N ≥ d (cid:16) ξ Nτ Nk , ξ Nτ Nk − (cid:17) ≤ N sup p d ( p, ζ N ) . (4.58)We estimate the last term: d ( p, ζ N ) ≤ sup Y ∈ D p M d (cid:16) p, exp p (cid:16) Y − µ p √ N + µ p N (cid:17)(cid:17) (4.59) ≤ sup Y ∈ D p M C √ N · F (cid:16) Y − (cid:16) − √ N (cid:17) µ p (cid:17) (4.60) ≤ C ( C + 1) . (4.61)Thus we have for k ≥ d (cid:16) ξ N , ξ Nτ Nk (cid:17) ≤ k (cid:16) C ( C + 1) (cid:17) . (4.62)Now for p ∈ M , ε >
0, and k ε defined as in Equation (4.56), consider the closed ball K p ( ε ) := { q ∈ M : d ( p, q ) ≤ R ( ε, T ) } , (4.63)with radius R ( ε, T ) = k ε (1 + C ( C + 1)) + 1 . (4.64)Then K p ( ε ) is closed and forward bounded, hence it is compact by Hopf–Rinow theorem.Eventually we get that ∀ p ∈ M and N ≥ P p (cid:0) ξ Nt / ∈ K p ( ε ) for some t ≤ T (cid:1) ≤ P p (cid:0) τ Nk ε ≤ T (cid:1) + P p (cid:0) ξ Nt / ∈ K p ( ε ) for some t ≤ T, τ Nk ε > T (cid:1) ≤ ε, (4.65)since the last summand equals to zero by construction of the set K p ( ε ). This finishes the proofof compact containment condition.So far, we have proved the sequence { ξ N } satisfies both Aldous criteria and the compactcontainment condition. Thus this sequence is tight. It is well known tightness implies beingrelatively compact. Thus any subsequence of { ξ N } has a further subsequence converging weaklyto some process ξ on M .We close this section by showing any limit process of { ξ N } has continuous paths almost surely. Proposition 4.10.
Any limit point ξ of geodesic random walks { ξ N } is a.s. continuous.Proof. The uniform elliptic condition implies that the jump sizes of the geodesic random walks ξ N converge to zero uniformly as N → ∞ , since d ( ξ Nt − , ξ Nt ) ≤ C + 1 √ N , ∀ t ∈ [0 , ∞ ) . (4.66)Hence the statement follows immediately from Theorem 3.10.2 of [14].28 .4 Convergence of geodesic random walks. In this section, we give the proof of Theorem 2.1. We already know the sequence { ξ N } isrelatively compact. To show the weak convergence, it remains to prove all limit points of { ξ N } have the same law. This is achieved by showing any limit point of this sequence is a solution toa well-posed martingale problem.We first need the following lemma. Recall that A defined in (2.8) is the limit of the generators A N . Lemma 4.11.
For any p ∈ M , any limit point ξ of { ξ N } and any f ∈ C ∞ K , we have f ( ξ t ) − f ( p ) − (cid:90) t Af ( ξ s ) d s, ∀ t ≥ , (4.67) is a martingale.Proof. It suffices to show that for any l ≥
1, any h , . . . , h l ∈ C b ( M ), any 0 ≤ s ≤ t , s , . . . , s l ∈ [ s, t ], and any f ∈ C ∞ K , the following holds. E (cid:104)(cid:16) f ( ξ t ) − f ( ξ s ) − (cid:90) ts Af ( ξ r ) d r (cid:17) l (cid:89) j =1 h j ( ξ s j ) (cid:105) = 0 . (4.68)Since ξ N is a Markov process for all N ≥
1, it follows for all 0 ≤ s ≤ t that f ( ξ Nt ) − f ( ξ Ns ) − (cid:90) ts A N f ( ξ Nr ) d r is a martingale. Hence for each N ≥ E (cid:104)(cid:16) f ( ξ Nt ) − f ( ξ Ns ) − (cid:90) ts A N f ( ξ Nr ) d r (cid:17) l (cid:89) j =1 h j ( ξ Ns j ) (cid:105) = 0 . (4.69)Separate the formula (4.68) into two terms, and let { ξ N k } be a subsequence converging weakly to ξ . Since ξ has continuous paths almost surely, the finite dimensional distributions of ξ N k alwaysconverge weakly to those of ξ (Theorem 3.7.8 of [14]). Thus we have E (cid:104)(cid:16) f ( ξ t ) − f ( ξ s ) (cid:17) l (cid:89) j =1 h j ( ξ s j ) (cid:105) = lim N k →∞ E (cid:104)(cid:16) f ( ξ N k t ) − f ( ξ N k s ) (cid:17) l (cid:89) j =1 h j ( ξ N k s j ) (cid:105) . (4.70)Furthermore, E (cid:104) (cid:90) ts A N k f ( ξ N k r ) d r · l (cid:89) j =1 h j (cid:16) ξ N k s j (cid:17) (cid:105) = E (cid:104) (cid:90) ts Af ( ξ N k r ) d r · l (cid:89) j =1 h j (cid:16) ξ N k s j (cid:17) (cid:105) + E (cid:104) (cid:90) ts ( A N k − A ) f ( ξ N k r ) d r · l (cid:89) j =1 h j (cid:16) ξ N k s j (cid:17) (cid:105) (4.71)and the latter summand vanishes as N k → ∞ because the functions h j are bounded and byProposition 4.3 lim N →∞ (cid:107) ( A N k − A ) f (cid:107) = 0 , ∀ f ∈ C ∞ K .
29o treat the first term, since x (cid:55)→ Af ( x ) is continuous and bounded, we have for each r ∈ [ s, t ]lim N k →∞ E (cid:104) Af ( ξ N k r ) · l (cid:89) j =1 h j (cid:16) ξ N k s j (cid:17) (cid:105) = E (cid:104) Af ( ξ r ) · l (cid:89) j =1 h j ( ξ s j ) (cid:105) . (4.72)Thus by Fubini’s and Lebesgue’s theorems we getlim N →∞ E (cid:104) (cid:90) ts Af ( ξ Nr ) d r · l (cid:89) j =1 h j ( ξ Ns j ) (cid:105) = (cid:90) ts lim N →∞ E (cid:104) Af ( ξ Nr ) · l (cid:89) j =1 h j ( ξ Ns j ) (cid:105) d r = E (cid:104) (cid:90) ts Af ( ξ r ) d r · l (cid:89) j =1 h j ( ξ s j ) (cid:105) . (4.73)and (4.68) is established. Proposition 4.12.
The martingale problem (4.67) has a unique solution which is stochasticallycomplete. Hence the sequence { ξ N } converges weakly.Proof. The well-posedness follows from the well-posedness of the martingale problem in R m .Indeed, in any chart U , the generator A is a second-order strongly elliptic operator with smoothcoefficients. We can extend the generator on U c such that its coefficients are uniformly Lipschitz.Then the martingale problem is well-posed, e.g. by Theorem 5.1.4 in [41]. By Theorem 4.6.1 of[14], the stopped martingale is also well posed for any initial distribution. The localized solutionsin countably many charts can glued together by Lemma 4.6.5 and Theorem 4.6.6 in [14], see alsoSection 4.11 in [25].In summary, we have shown the sequence { ξ N } converges weakly to some process ξ on M which is a solution to a well-posed martingale problem. This completes the proof of Theorem2.1.Eventually let us also prove, since it is an important and useful property, that the limit processis Feller. Proposition 4.13.
The limit process ξ is Feller, i.e. its semigroup preserves C ( M ) .Proof. In any chart, ξ is a non-degenerate diffusion with smooth coefficients, hence its semigroupmaps C ( M ) to C ( M ).Denote ( T t ) t ≥ the semigroup of ξ as usual. Let f ∈ C ( M ), t > ε > C ε such that | f ( x ) | ≤ ε for x / ∈ C ε . Define R ( ε, t ) as in (4.64) in Lemma 4.9. Bythis lemma, for any p ∈ M such that d ( p, C ε ) > R ( ε, t ), we have | E p f ( ξ t ) | ≤ E p | f ( ξ t ) | I ( τ R ≤ t ) + E p | f ( ξ t ) | I ( τ R > t ) ≤ (cid:107) f (cid:107) · P p ( τ R ≤ t ) + ε ≤ ( (cid:107) f (cid:107) + 1) ε. (4.74)Thus ( T t )( f ) vanishes at infinity for f ∈ C . As ξ is a limit point of { ξ N } , Lemma 4.7 impliessup p ∈ M P p ( d ( p, ξ t ) > δ ) ≤ C ( δ ) t, ∀ δ > , ∀ t ≥ . The strong continuity of the semigroup ( T t ) follows.30 eferences [1] P. L. Antonelli, A. B´ona, and M. A. Slawi´nski. Seismic rays as Finsler geodesics. NonlinearAnalysis: Real World Applications , 4(5):711–722, 2003.[2] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto.
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Tianyu MaInstitut f¨ur MathematikFriedrich–Schiller Universit¨at Jena07737 Jena, Germany [email protected]
Vladimir S. MatveevInstitut f¨ur MathematikFriedrich–Schiller Universit¨at Jena07737 Jena, Germany [email protected]
Ilya PavlyukevichInstitut f¨ur MathematikFriedrich–Schiller Universit¨at Jena07737 Jena, Germany [email protected]@uni-jena.de