Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds

We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

Let us recall one of the first constructions of a random walk which is due to K. Pearson in 1905 [33]. A more physically motivated approach is in the paper [13] of A. Einstein from the same year.
We start from a point p ∈ R 2 , choose a random direction at the tangent space, go for distance 1 along the straight line starting at this direction, and then repeat the procedure iteratively. We obtain a stochastic process whose trajectories are piecewise-linear curves, see Fig. 1. 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 procedure of choosing the random direction is invariant with respect to the isometry group of the flat R 2 which was the case in [13,33], then by the Functional Central Limit Theorem the limit of this sequence as N → ∞ exists and is 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 random direction at a current position p (i.e., a probability distribution ν p on the space of tangent vectors at the point p) and an analogue of the notion of a straight line, which describes the motion of a small particle with no external forces acting upon it.
In many systems in physics and natural sciences, small particles with no external forces acting upon them move along geodesics of a Finsler metric. We give necessary definitions in §2.1. Recall that geodesics are smooth curves and, similar to the straight lines, the initial point and the initial velocity vector determine the geodesic. The above definition of the random walk is immediately generalised to this case. Indeed, starting from a point p of a Finsler manifold (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 geodesic starting at p with the initial velocity v and then repeat the procedure (if ν p is not centered we rescale it as in §2.2). We obtain a stochastic process whose trajectories are piecewise geodesic curves (e.g., they are glued together from geodesic segments).
The present paper studies such geodesic random walks on Finsler manifolds and their limit diffusion processes, and concentrates on the fundamental question of the existence and uniqueness of the limit process. Our main result is that under assumptions natural from the viewpoint of Finsler geometry, the limit process exists and is unique. Moreover, it is a diffusion process whose generator is a non-degenerate elliptic second order partial differential operator for which we give a precise formula.
The Riemannian version of our result (recall that Riemannian metrics are Finsler metrics) was obtained e.g. by E. Jørgensen [22].
Geodesic random walks on Finsler manifolds and their limit processes are of course natural topics from the viewpoint of both differential geometry and theory of stochastic processes. They may have applied interest since Finsler manifolds are used to model different physical situations with anisotropies at an infinitesimal level, see e.g. [2,1,10,12,17,20,28,34,45], and may also be used for certain models in information geometry, see e.g. [40].
Although Brownian motions and diffusion processes on Finsler manifolds were discussed in the literature (see e.g. the books [4,44]), the very basic question of the existence and uniqueness of the limit process for Finsler geodesic random walks has not been rigorously treated.
More precisely, the work [44] on Finsler Brownian motions goes in the other direction: It starts from a stochastic differential equation which is constructed by a Finsler metric F , a volume form µ, and an extra data u 0 ∈ H 1 0 (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 sequence of geodesic random walks.
This general approach, in which one starts with an elliptic differential operator (or a Dirichlet form) in order to construct a diffusion process, is a very popular and powerful approach to diffusion processes on metric spaces. It allows in particular to treat the case of non-smooth background metric structures, see e.g. [18,27,42]. This approach does not ensure that the resulting stochastic process is the limit process of a sequence of random walks. If the background is almost Riemannian (say, Alexandrov with bounded curvature, as in [18] and [27]), the best one can do is to relate random walks on the Riemannian spaces approximating our metric space to the diffusion process on our metric space. These results cannot be applied in the Finslerian situation, since Finsler metrics cannot be approximated by Riemannian metrics. Our results will possibly allow to extend this group of methods to a Finslerian situation and we plan to do this in our future works.
Let us now discuss the corresponding results of the book [4], where many different approaches of constructing different non-equivalent diffusion processes (on the manifold or on the tangent bundle 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 Finsler metric). Unfortunately no rigorous proof of convergence is given: it is merely claimed that the limit 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 Riemannian manifolds. The methods discussed there assume and rely on the special form of the probability measure ν p on tangent spaces. Moreover, it is assumed that the Riemannian manifold is stochastically complete. The property of stochastic completeness is a nontrivial property, and examples show that not all complete manifolds are stochastically complete. In the Riemannian case, there is a number of criteria of stochastic completeness, see e.g. [19,46]. In particular, if the Ricci curvature of a complete Riemannian manifold is bounded from below, the manifold is stochastically complete. In the Finslerian situation, we did not find any relevant works on stochastic completeness and the claim of [4, §A2] that the methods of [35,36] can easily be applied in the Finslerian situation looks overoptimistic.
Note that as a by-product, we have proved that every complete Finsler manifold of bounded geometry (see Definition 2.1) is stochastically complete; that is, the limit process of Finsler geodesic random walks is stochastically complete in the sense of [21, §4.2]. It is interesting to try to relax the assumption of bounded geometry in this statement and we plan to do this in future works.
A very successful approach to geodesic random walks and diffusion processes on Riemannian manifolds, which allows essential freedom in the choice of the probability measures ν p , is in [22]. Many arguments in [22] are based on the following property which holds in the Riemannian but not in the Finslerian case: Consider an arc-length parametrized geodesic segment γ : [0, ε] → M of a (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 geodesic geodesics γ v and γ vε 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 vε (t), behaves, for ε → 0 and t → 0, as ε(1 + Ct 2 ). In the Euclidean case, the distance does not depend on t at all and is equal to ε. In the Finslerian situation, this property does not hold for a generic metric and a straightforward generalisation of [22] is not possible.
In this paper we prove that under the assumptions natural from the viewpoint of Finsler geometry (everything is smooth, the manifold is complete and has bounded geometry), the sequence of 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 integral formula for its coefficients.
As explained above, the generator of the limit diffusion process is a non-degenerate elliptic operator. If the probability measure ν p on each T p M is constructed by F |TpM (we give examples in §2.3.2) then this elliptic operator is a natural candidate for a Beltrami-Laplace operator of the Finsler metric. Note that, different from the Riemannian case, there exist many different Finslerian analogues of the Beltrami-Laplace operator. We refer to [3], where many different constructions of the Riemannian Beltrami-Laplace operator are mimicked in the Finslerian setting. In the Riemannian case they all give the same Beltrami-Laplace operator. In the Finslerian case one obtains different operators. Most operators in [3] are linear but there also exist nonlinear versions of the Finslerian Betrami-Laplace operators, see e.g. [32,37].
An interesting by-product of our result is that the generator of the limit diffusion process corresponding to Finsler geodesic random walks coincides, up to first order terms (the so-called "drift"), with that of a Riemannian Brownian motion. This result of us explains why it is hard or even impossible to experimentally distinguish a diffusion process coming from a Riemannian metric from that of coming from a Finsler metric. See §2.3.1 for more details.
Naturally, the topic of this paper, and therefore also the methods of the proof, belong both to differential geometry and to the theory of stochastic processes. The group of the methods coming from stochastic processes is actually standard for this type of problems (though nontrivial) and was understood at least in the 70th-80th. The novelty which allowed to solve this natural and actively attacked problem came from Finsler geometry, and the key lemma is Lemma 4.5, whose proof uses a nontrivial and not widely known result of [38, §15].
2 Setting and results.

Finsler manifolds.
First we recall the basic definitions in Finsler geometry. Let M := M m be a m-dimensional manifold, m ≥ 1. Suppose that (x 1 , . . . , x m ) is a local coordinate at some p ∈ M . Then y i = ∂x i induces a local coordinate (x 1 , . . . , x m , y 1 , . . . , 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 function F : T M → R ≥0 called the Finsler metric (Finsler function) satisfying the following conditions: Positive Homogeneity: For any (x, y) ∈ T x M and λ ≥ 0, we have F (x, λy) = λF (x, y).
Strong Convexity: For 0 = (x, y) ∈ T x M , the fundamental tensor defined by is strictly positive definite.
The indicatrix bundle of (M, F ) is defined by For any p ∈ M , the fibre I p M of IM is a convex hypersurface in T p M diffeomorphic to S m−1 .
If (M, g) is a Riemannian manifold, one can naturally endow it with a Finsler metric by setting F (Y ) := g(Y, Y ), Y ∈ T M . Conversely, a Finsler function corresponds to some Riemannian metric 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 Finslerian situation. A smooth curve γ : [a, b] → M is a geodesic, if it is a stationary point of the energy functional 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 anḋ γ(0) = Y . We define the exponential map at p to be for all Y ∈ T p M such that γ Y (t) is defined for t ∈ [0, 1]. We say (M, F ) is forward complete if for 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 The Finsler function F defines the following asymmetric and symmetrized distances on M : Like in the Riemannian case, geodesics of Finsler metrics are local distance minimizing (with respect to d a ) curves. The formula (2.2) ensures that they are parametrised proportional to the arc-length parameter. Note, as F is in general not reversible, i.e. F (x, y) ≡ F (x, −y), the distance function 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 our Finsler manifold are uniformly bounded. The flag curvature K can be thought as a generalisation of the Riemannian sectional curvature. The definition of T -curvature (see [38, §10.1]) is essentially Finslerian since it vanishes for Riemannian manifolds.
Within the whole paper we assume the following set of hypotheses.
H c : The manifold (M, F ) is connected and forward complete.
The manifold (M, F ) has bounded geometry in the following sense: We say a Finsler manifold (M, F ) has bounded geometry if the followings hold: 1. Uniform ellipticity: There is some constant C > 1 such that for any p ∈ M and any non-zero u, v ∈ T p M , we have 2. The flag curvature K is bounded uniformly and absolutely by some constant λ > 0, namely K ≤ 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 Note that all objects used in the definition of "bounded geometry" are microlocal, and for an explicitly 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 We denote the unit discs on T M by Let d be the symmetrized distance defined by (2.5).
We denote the open balls by: In this paper B is the space of Borel measurable real valued functions on M , C 0 is the space of continuous that vanish at infinity, C ∞ is the space of smooth functions, C ∞ K is the space of smooth functions with compact support.

Rescaled geodesic random walks.
As a motivation for "rescaling", let us consider the following example of a geodesic random walk: The manifold is R with the standard flat metric and ν p is defined as follows. At every point p, the support of ν p is two vectors {1, −1} ⊂ T p R = R such that the probability of −1 is 1/4, and probability of 1 is 3/4. That is, the particle goes the distance 1/ √ N with probability 1/4 in the negative direction and with probability 3/4 in the positive direction.
In this case the mean value of the position of the particle in one step is 1/(2 √ N ) in the positive direction, so in N steps the mean value is √ N /2. For N → ∞, most trajectories "escape to 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 is not a rectifiable curve).
Because of this, we introduce below the rescaled random walk (we will formalize this definition in §3.2). We denote the mean of ν p by µ p , and modify the measure ν p by shifting it in T p M by the vector −µ p + µ p / √ N , i.e.ν p (y) := ν p (y − µ p + µ p / √ N ). We will call this operation the rescaling of measure.
Let us explain this operation. First we note that the easiest would be to shift the measure by the vector −µ p . This will make the mean of the new measure equal to 0, and the phenomenon demonstrated in the example above does not appear. Note that some papers on random walks on Finsler metric, for example [4], assume that both Finsler metric and the measure µ p are centrallysymmetric on every T p M (the so-called reversible situation). We do not want to do it since most examples 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 the random 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 physical model, the mean of ν p should somehow come in the definition. Of course, it may come with any other 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 shifting it by βµ p , then the rescaled measure will be shifted by (1 + β)µ p / √ N . Thus, all results of our paper can be applied for any α, in particular for α = 0.
We also expect that such rescaling of the measure is physically-relevant, since microscopic particles can not make too long jumps because of friction and collisions; so even if the probability of the particle to go to the "right" is higher than the probability to go to the "left", the particle does not escape to infinity in short time, contrary to what is suggested by the random walk described in the beginning of this section.

The main result.
We will assume that the Finsler manifold (M, F ) is connected and forward complete (Hypothesis H c ) and has bounded geometry (Hypothesis H b ). In addition, we make the following assumption on the family of measures {ν p }.
H ν : We assume that ν = {ν p } is a smooth family of probability measures inside DM : In the first case we require that ν is a smooth m-form on DM such that for every p, the restriction ν p := ν| TpM is a form on the disc probability measure. Similarly, in the second case ν is a smooth (m − 1)-form on IM such that for each p ∈ M the restriction ν p := ν| IpM is a probability measure.
This hypothesis is very natural from the viewpoint of Finsler geometry, and covers many choices that have their natural counterparts in the Riemannian setting; we will give a few examples in §2. 3

.2.
Our main result is the following theorem: Let Hypotheses H c , H b and H ν be satisfied. Consider a family of geodesic random walks starting at p 0 constructed from (M, F ) and {ν p } p∈M . Then, this sequence has a unique weak limit ξ. The process ξ is a diffusion whose generator is a non-degenerate elliptic differential operator A with smooth coefficients given by Here γ Y −µp is the geodesic with initial vector Y − µ p . In the local coordinates, it has the following form: where Γ k ij are the formal Christoffel symbols of the second kind given by x i x j f . Moreover, ξ is stochastically complete.

Limit diffusion as a Riemannian Brownian motion with drift.
Recall that the Riemannian Brownian motion is a diffusion process which is the limit of geodesic random walks with identically distributed steps. Here identically distributed should be understood as follows: the probability measure ν p is invariant with respect to the parallel transport 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 the inverse of the Riemannian metric.
By Theorem 2.1, in the Finslerian case the generator A of the limit process of the geodesic random walk is a second order non-degenerate elliptic differential operator on M . Hence the symbol 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 of generators may be different). This should be the reason why Finslerian effects related to diffusion were not observed experimentally in physical or natural science systems, even in those where the free motion of particles corresponds to geodesics of a certain Finsler metric. See e.g. [15] where in a highly anisotropic situation (diffusion weighted magnetic resonance imaging of brain), the measurement 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 the model (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 the system in question or determining their parameters.

Canonical constructions of Riemannian metrics.
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 construction implies that the metric is invariant under the group SO(g), which implies that in the orthogonal coordinates (y 1 , . . . , y m ) on T p M it is given by φ((y 1 ) 2 + · · · + (y m ) 2 ) dy 1 ∧ · · · ∧ dy m . The function φ is the same for all points p, has the property that it is nonnegative and that the integral R n φ((y 1 ) 2 + · · · + (y m ) 2 ) dy 1 ∧ · · · ∧ dy m = 1. In the Finslerian situations there are many natural non-equivalent constructions of a measure on T p M . Let us recall the following three.
Measure coming from the Lebesgue measure: For any p ∈ M , let ω p = dy 1 ∧ · · · ∧ dy m be a Lebesgue measure on T p M . It is known that it is unique up to a positive coefficient. We restrict it to the ball D p M (that is, the measure of an open set U ⊂ T p M it the Lebesguemeasure of the intersection D p M ∩ U ), and normalize it such that it becomes a probability measure.
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 volume on I p M induced by g ij we obtain probability measure .
This probability measure is close to the one used in [4, §A2].
Measure coming from the Hilbert form: Denote P + (M ) the positive projectivized tangent bundle. The Hilbert 1-formω = F y i dx i defined on T M \ {0} is actually a pull back of some 1-form ω on P + (M ) by the standard projection. It is well known that ω ∧ (dω) m−1 defines a volume form on P + (M ) IM . Let i p : I p M → IM be the standard inclusion and π : IM → M be the canonical projection. It is known (see e.g. [7]) that there exists a (m − 1)-form α F on IM and a volume form ω F on M such that α F | IpM is a unique volume form on I p M for each p ∈ M with vol α F (I p M ) = 1, (2.11) Hence we can take ν p := vol α F on I p M .
Each of these measures satisfies the Hypothesis H ν , and is coordinate-independently constructed from F . In the case the Finsler metric is reversible, the dual of the symbol of the generator 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 modification of 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 the Finsler Laplacian from [7] appear to be effective tools for solving different problems in Finsler geometry; we expect that other natural choices of the measure ν p may also be useful in Finsler geometry.

Example: Limit diffusion for a Katok Finsler metric.
Let (S 2 , g) be the unit sphere endowed with the standard Riemannian metric. Katok metric is constructed as follows. Let X be the vector field of rotation around the axis connecting the north and south poles of the sphere such that g(X, X) < 1. In the the following spherical coordinate on S 2 , (ψ, θ) → (cos(ψ) cos(θ), sin(ψ) cos(θ), sin(θ)). (2.12) We have Now for any p ∈ M , the indicatrix I p M of the constructed Finsler function F is obtained by shifting the unit sphere S p S 2 of g (which is the indicatrix with respect to g) by X. That is, This yields a well-defined Finsler metric as g(X, X) < 1. This family of metrics depending on the parameter r was constructed by A. Katok in [24]. It is widely used in Finsler geometry and in the theory of dynamical system as source of examples and counterexamples. It has constant flag curvature by [6,16,39], and by [9], any metric of constant flag curvature on the 2-sphere has geodesic flow conjugate to that of a Katok metric.
As the measure ν p we consider the Lebesgue measure as described in §2.3.2; let us calculate the generator of the corresponding diffusion process ξ.
By Theorem 2.1, the diffusion process ξ generated by {ν p } p∈M has generator A such that Here Γ k ij 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 R m , we also denote this Lebesgue measure by ν p for simplicity. For any p ∈ M , the set 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 2 such that for any integrable second order homogeneous function h on T p M , we have Because ν p is invariant under any orthogonal transformation on T p S 2 with respect to g, it is clear η p is a multiple of the canonical angular measure m p on S p S 2 with respect to g. A straight forward computation shows η p = 1 4π m p . Hence from (2.14), we get Let ∆ be the Beltrami-Laplace operator of g, and letΓ k ij be the Christoffel symbols of g. A straightforward computation yields This implies 1 8 ∆ and A have the same symbol. To compute the drift of A, we assume without loss of generality that 0 ≤ r < 1. First we have Let Φ t be the flow generated by X, we know from [16, Theorem 1] that if γ(t) is a geodesic of (S 2 , g) with g(γ,γ) = c, thenγ(t) = Φ ct • γ(t) is a geodesic of F with initial vectorγ (0) = γ(0) + cX(γ(0)). But in the spherical coordinate given (2.12), the flow Φ t simply has the form Φ t (ψ, θ) = (ψ + rt, θ).
Then in this coordinate, we have By the geodesic equation, this is equivalent to This is the drift of the generator A.
On Figure 2 one clearly sees the difference in the behaviours of the Brownian motion of the initial round metric on S 2 and of the diffusion process corresponding to the Katok metric with r = 1/2 due to the drift given in (2.18). Of course the pictures are just the pictures of the corresponding geodesic random walks with a sufficiently large N . Note that the same random seed was used in both pictures.

Preliminaries.
In this section, we give a short review of the tools in Finsler geometry which will be used in our proof in later sections and formalise definitions of random geodesic walks which will allow us to apply the machinery from the theory of stochastic processes.

Finsler geodesics and properties of bounded geometry.
It is well-known that stationary points of the energy functional (2.2) are solutions of the Euler-Lagrange equation which in our situation is equivalent to the following system of ODEs: Here Γ k ij are the formal Christoffel symbols of the 2nd kind: It is immediate from (3.3) that for any λ > 0 and y = 0, we have Denote the class of real-valued k-times continuously differentiable real-valued functions on M with compact support by C k is the geodesic with initial vector Y as before.

5)
wherever it is well defined.
Proof. First we show there is some constant c such that . LetX be the geodesic spray on T M , and denote π : T M → M the canonical projection. The function π * f is For any p ∈ M and Y ∈ T p M , we have For p / ∈ K f , we have f vanishes identically on some neighbourhood of p. Then the function Next, given any geodesic γ Y , let Y (t) be its velocity field. We have

Formal definition of rescaled geodesic random walks
We begin this section by a brief review of the basic definitions in Markov processes used in this paper. Roughly speaking, a stochastic process is said to be Markovian if its future states depend only upon the present state, regardless of its past state. , for all n ≥ 1, 0 ≤ s 1 < · · · < s n < s < t we have, P(ξ t ∈ B|ξ s 1 , . . . , ξ sn , 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, ∀0 ≤ s ≤ t.
We say (ξ t ) t≥0 is time homogeneous if the following holds.
All Markov processes considered in this paper are time-homogeneous. A time homogenous Markov process ξ defines a semigroup T = (T t ) t≥0 of linear operators on the measurable functions B on M by We say a Markov process ξ is Feller if the semigroup T is a strongly continuous semigroup of positive contractions on the Banach space C 0 .
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 measures such that each ν p is a probability measure on T p M . Denote by µ p the mean of ν p µ p := TpM Y ν p (dY ).
In our setting (Hypothesis H ν ), the probability measures are compactly supported so µ p exists and is finite. Definition 3.2. Let N ≥ 1 and let p 0 ∈ M be fixed. A random process (ζ N k , Y N k+1 ) k≥0 is called a (rescaled) discrete time geodesic random walk on M with initial point p 0 and with increments for any measurable B ⊆ T p M , Hence the processes ζ N are defined by the family of measures {ν p } p∈M and the geometry of 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 N k+1 depend only on the current position ζ N k and not on the previous positions and increments. More precisely, we introduce the natural filtration and say that the increments of ζ N are independent if for each It is clear that ζ N is a homogeneous discrete time M -valued Markov chain with the one-step transition operator Since we work in a continuous time setting it is convenient to transform the discrete time Markov chain ζ N into a continuous time Markov process. This can be done by a standard subordination procedure.
Let Q = (Q t ) t≥0 be a standard Poisson process independent of {ζ N }. Define a pseudo-Poisson process 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 the transition semigroup T N = (T N t ) t≥0 of ξ N t has the form Finally we introduce a of family continuous M -valued processes defined bŷ Since the manifold is geodesically complete, the processes ζ N , ξ N andξ N are well-defined for each N ≥ 1. By construction the processesξ N have piecewise smooth sample paths consisting of geodesic segments, and induce probability distributionsP N on the path space C([0, ∞), M ) of continuous M -valued functions. These are the geodesic random walks introduced and discussed in the Introduction, see Fig. 1 there, and in §2.4, see Fig. 2. Although the processesξ N are not Markovian, the convergence of the continuous time processes (ζ N [N t] ) t≥0 , (ξ N t ) t≥0 and (ξ N t ) t≥0 is equivalent, see, e.g. [23,Theorem 17.28]. In the next sections we will mainly work with the Markov processes ξ N .

Proof of the main theorem.
In this section we prove the convergence of the geodesic random walks {ξ N }. We will assume that the Finsler manifold (M, F ) is forward complete and connected (Hypothesis H c ) and has bounded geometry (Hypothesis H b ) and that the measures {ν p } p∈M satisfy the condition H ν from §2.3.

Generators of geodesic random walks.
In this section we show the N -scaled geodesic random walks on a complete Finsler manifold (M, F ) with bounded geometry are Feller. is also C k -smooth.
This lemma is obvious since each ν p is only supported on D p M . Indeed, integral over a compact 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 the generators.
Proof. Let N ≥ 1 be fixed. Since by constriction T N is a strongly continuous semigroup of a pseudo-Poisson process, its generator has the form (4.2) by Theorem 19.2 from [23]. It is conservative due to assumption H c . Let us show that T N t maps C 0 into itself for t ≥ 0. Since we have the series in (3.11) converges uniformly. It suffices to show P N maps C 0 into itself. By Lemma 4.1 the mean value µ p is a C ∞ vector field. Since the exponential map for Finsler manifold is at least C 1 , then P N maps continuous functions into continuous functions.
For any ε > 0, choose some compact K ⊆ M such that |f (x)| < ε 2 for x / ∈ K. Fix any p 0 ∈ K, and define the closed forward balls at p 0 for R ≥ 0 by By the Hopf-Rinow theorem (see Theorem 6.6.1 of [5]), the forward closed balls B + p 0 (R) are also compact.
For any 0 = Y ∈ T p M and p ∈ M , the uniform ellipticity condition in Definition 2.1 gives It follows that Let R 1 := C(C + 2 + CR 0 ), then ∀p ∈ (B + p 0 (R 1 )) c and ∀q ∈ K, we have On the other hand, for p ∈ M and Y ∈ D p M , we have Hence, ∀Y ∈ D p M and p ∈ (B + p (R 1 )) c , we have e N p (Y ) / ∈ K. It follows that ∀p ∈ (B + p (R 1 )) c : That is to say P N f ≤ ε 2 outside the compact set B + P (R 1 ). We conclude that P N f ∈ C 0 . This completes the proof.

Convergence of the generators of geodesic random walks.
In this section, we prove the generators A N converge on the space C ∞ K to some second order elliptic operator with smooth coefficients. Denote Then A is a second order positive definite elliptic operator of smooth coefficients and for each Proof. The proof follows the steps from [22] in the Riemannian case. By computing the Taylor expansion of A N f , we show the convergence of the first and second order terms and vanishing of other higher order terms as N → ∞. Take any f ∈ C ∞ K . We have Then for any p ∈ M and Y ∈ D p M , the Taylor expansion of gives

Thus we have
(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 > 0 such that The last term in (4.15) tends to zero, since ν p is only supported on D p M .
For the second order term, in a canonical coordinate of T M , we have (4.18) Since the formal Christoffel symbols are bounded on each compact local coordinate, the righthand-side of (4.18) converges to Choose a smooth coordinate on some open U ⊆ M . The chain rule implies A has the following form in this coordinate.
Because f has compact support, we have It follows that the symbol of A is:   Proof. The statement follows from the Aldous criteria (Lemma 4.8) and the compact containment condition (Lemma (4.9)) that will be proven in this section.
Our 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 bound inj M can be zero. Since the non-symmetrized distance function d a (p, ·) is smooth only within the injective radius, we work on the tangent bundle T M to bypass this technical problem. To prepare the proof of Lemma 4.8 as well as Lemma 4.9, for R ≥ 0, define We make the following construction.
The condition K ≤ λ implies there exists some 0 < δ c < 1 so that the conjugate radius con M > δ c , see Proposition 3.2. For each p ∈ M , the exponential map exp p is a smooth immersion on D p (δ c ) except possibly at 0. Then we can construct a geodesically complete smooth Finsler function F p on T p M such that F p = (exp p ) * (F ) on D p ( δc 2 ), while F p is the standard Minkowski metric on T p M \ D p (1), under any standard identification T p M R m . To distinguish it from the distance functions on (M, F ), we denote the asymmetric and symmetric distance on (T p M, F p ) by d p a and d p , respectively. Note that the injective radius of F p at 0 ∈ T p M is at least δc 4 . Now for each p ∈ M we construct the measures {ν q } q∈TpM , so that on D p (δ c /2), the measures {ν q } q∈Dp(δc/2) are the lift of {ν o } o∈M by the exponential map exp p . In addition, we require the measures {ν q } q∈TpM 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 the Finsler manifold (T p M, F p ) starting at 0 ∈ T p M , using the prescribed measures {ν q } q∈TpM as in Section 3.2. Note as (T p M, F p ) satisfies H b and H c , all results we proved earlier are true for the random walks ξ N,p . Lemma 4.5. There exists some δ 0 > 0 so that for each δ ∈ (0, δ 0 ), there exists a family of functions {f δ p } p∈M such that 1. Each f δ p is a function on T p M with 0 ≤ f δ p ≤ 1 such that f δ p (0) = 1 and f δ p (q) = 0 if q / ∈ D p (δ).

Denote
is smooth on T p M and satisfies condition 1.
To prove condition 2, for any q ∈ (T p M, F p ), letμ q be the mean ofν q . An argument similar to Proposition 4.2 shows that for any q ∈ T p M By Taylor theorem, there exist functions {t N } with Using Equations (4.23) and (4.27), we get We need to show the equation above is uniformly bounded for all p ∈ M , q ∈ T p M and N ≥ 1. First we show for each 0 < δ < δ 0 , For any q ∈ T p M such that δ 4 ≤ d a (0, q) ≤ δ 2 , the Finlser metric F p | Tq(TpM ) and the measureν q are the pull backs of F and {ν o } o∈M by exp p , respectively. Thus F p (μ q ) < 1 and F p (−μ q ) ≤ C. It follows that The function d p a (0, ·) is smooth at q For q ∈ T p M with δ 4 ≤ d a (0, q) ≤ δ 2 . Hence Equation (4.29) holds by the chain rule. Now 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 we have For the case 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 p a (0, ·) the Finsler gradient, see, e.g. [38,Equation (3.14) Hereg 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 uniform elliptic conditions for any 0 = u, v ∈ T q (T p M ) with q ∈ D p (δ). It follows that for all 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, because exp p restricted to (D p (δ), F p ) is an isometric immersion. Since the injective radius of F p at 0 is at least δc 4 > δ, the Hessian comparison theorem implies: Using the factĝ(∇ρ, ∇ρ) = F 2 p (∇ρ) = 1 on D p (δ) \ {0}, we get If x ∈ D p (δ), for any tangent vectors Y 1 , Y 2 ∈ T x (T p M ) with Y 1 = 0, the fundamental inequality in Finsler geometry (see 1.2.16 of [5]) and the inequality Substituting this into (4.36) and using uniform ellipticity, for Y such that F p (Y ) ≤ 1 and (4.32) holds, we obtainĝ Then for all N ≥ 1 and Y with F p (Y ) ≤ 1, we have the estimate: This shows the second derivative of h N (Y )(t) evaluated at t = t N (Y ) is also uniformly and absolutely bounded, if (4.32) holds true. Then there exists someC(δ) > 0 such that condition 2 holds. This completes the proof.
The family of functions {f δ p } will be used now to estimate the first exit time from a δ-balls of the geodesic random walk ξ N .
For each p ∈ M , N ≥ 1 and δ > 0, define the following stopping times for the random walk ξ N on M and ξ N,p on T p M : Lemma 4.6. For any p ∈ M , N ≥ 1 and δ such that 0 < δ < δ 0 and 2(C+1) √ N < δc 4 , we have Proof. The geodesic random walks ξ N,p and ξ N are constructed by randomizing the time of the discrete Markov processes ζ N,p and ζ N using a Poisson process, respectively. Hence it suffices to show for each pair (N, δ) satisfies the condition in the lemma, the following holds For r > 0, define the closed δ-balls of the symmetrized distances on T p M and M respectively: Because δ < δ 0 < δc 4(C+1) , we have B p 0 (δ) ⊂ D p (δ c /2). Hence exp p maps (B p 0 (δ), F p ) inside (B p (δ), F ) by an isometric immersion. Now for each k ≥ 0 define the following Borel subprobabilty measures on B p 0 (δ) and B p (δ), respectively.
Hence for any q ∈ B p 0 (δ) and 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 0 (δ), set q 1 = exp p (q). Sinceν q is the pull-back of ν q 1 by (d exp p (q)), then (4.46) implies for any Borel setÊ ⊂ B p 0 (δ), we have The inequality in the previous formula appears because the exponential map exp p restricted to B p 0 (δ) is not necessarily injective. In particular, for any Borel E ⊂ B p (δ), we get By the Markov property, we have for all k ≥ 1 Hence we have the following chain of inequalities: where the first inequality comes for the induction assumption, and the second inequality is due to (4.48).
In particular, for all k ≥ 0, the inequality 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 .
In particular, Proof. We adapt ideas from [26] by Kunita. The proof is divided into two steps. First we consider the exit times for the lifted random walks ξ N,p , and we show where 0 ∈ T p M andC(δ) > 0 is the constant defined in Lemma 4.5. Next we use Lemma 4.6 and (4.51) to prove (4.49), and (4.50) directly follows from (4.49).
Note that due to (4.49), we have P p (τ N,δ ≤ t * ) ≤ C(δ)t * ≤ 1 2 . Hence we obtain This proves the second inequality of the lemma.
Now we are ready to show the Aldous criteria hold in our situation. Proof. Let δ, s > 0 be fixed. For each N ≥ 1, p ∈ M , h ∈ [0, s] and a stopping time τ we have The strong Markov property of ξ N yields Thus by Lemma 4.7, we have 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 > 0, T ≥ 0 and p ∈ M there is a compact neighborhood K ε (p) ⊆ M of p such that Proof. Let us define the following sequence of exit times (as usual, we set inf ∅ = +∞). By Lemma 4.7, there exists some constant c 1 ∈ (0, 1) such that Then for k ≥ 1 and ∀N ≥ 1, the strong Markov property yields For any ε > 0 and T ≥ 0, define Then the exponential Markov inequality gives ∀p ∈ M , ∀N ≥ 1: (4.57) By construction and the triangle inequality we have that for each k ≥ 1, each N ≥ 1 We estimate the last term: 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 ≥ 1: P p ξ N t / ∈ K p (ε) for some t ≤ T ≤ P p τ N kε ≤ T + P p ξ N t / ∈ K p (ε) for some t ≤ T, τ N kε > T ≤ ε, (4.65) since the last summand equals to zero by construction of the set K p (ε). This finishes the proof of compact containment condition.
So far, we have proved the sequence {ξ N } satisfies both Aldous criteria and the compact containment condition. Thus this sequence is tight. It is well known tightness implies being relatively compact. Thus any subsequence of {ξ N } has a further subsequence converging weakly to some process ξ on M .
We close this section by showing any limit process of {ξ N } has continuous paths almost surely.

Convergence of geodesic random walks.
In this section, we give the proof of Theorem 2.1. We already know the sequence {ξ N } is relatively 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 to a 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 . Af (ξ s ) ds, ∀t ≥ 0, (4.67) is a martingale.
Proof. It suffices to show that for any l ≥ 1, any h 1 , . . . , h l ∈ C b (M ), any 0 ≤ s ≤ t, s 1 , . . . , s l ∈ [s, t], and any f ∈ C ∞ K , the following holds. 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 always converge weakly to those of ξ (Theorem 3.7.8 of [14]). Thus we have Furthermore, To treat the first term, since x → Af (x) is continuous and bounded, we have for each r ∈ [s, t] h j (ξ s j ) .   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 smooth coefficients. 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 solutions in countably many charts can glued together by Lemma 4.6.5 and Theorem 4.6.6 in [14], see also Section 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 Theorem 2.1.
Eventually let us also prove, since it is an important and useful property, that the limit process is Feller.
Proof. In any chart, ξ is a non-degenerate diffusion with smooth coefficients, hence its semigroup maps C 0 (M ) to C(M ).
The strong continuity of the semigroup (T t ) follows.