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

Tianyu Ma, Vladimir S. Matveev and Ilya Pavlyukevich

###### Abstract

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.
MSC 2000: 53B40, 53C60, 82B41, 82C41
Key words: Geodesic random walks, weak convergence, Finsler manifold,
diffusion process, Riemannian Brownian motion, averaged metric, bounded
geometry.

###### Contents

  1. 1 Introduction.
  2. 2 Setting and results.
    1. 2.1 Finsler manifolds.
    2. 2.2 Rescaled geodesic random walks.
    3. 2.3 The main result.
      1. 2.3.1 Limit diffusion as a Riemannian Brownian motion with drift.
      2. 2.3.2 Canonical constructions of Riemannian metrics.
    4. 2.4 Example: Limit diffusion for a Katok Finsler metric.
  3. 3 Preliminaries.
    1. 3.1 Finsler geodesics and properties of bounded geometry.
    2. 3.2 Formal definition of rescaled geodesic random walks
  4. 4 Proof of the main theorem.
    1. 4.1 Generators of geodesic random walks.
    2. 4.2 Convergence of the generators of geodesic random walks.
    3. 4.3 Tightness of the family $\\{\xi^{N}\\}$.
    4. 4.4 Convergence of geodesic random walks.

##  1 Introduction.

Many processes in physics and natural sciences can be described with the help
of random walks and their limit processes, the so-called diffusion processes.
A possible philosophical explanation of this experimentally observed
phenomenon is that the limit of random walks reflects the microscopic nature
of the situation: Even fully deterministic microscopic systems can give rise
to erratic seemingly random motions, practically indistinguishable from those
produced by a stochastic process.

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 $p\in\mathbb{R}^{2}$, 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.

Figure 1: 50 steps of a Pearson random walk.

It is natural to renormalise this process as follows: we assume that the steps
have length not $1$ but $1/\sqrt{N}$ 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 $\mathbb{R}^{2}$ which was the case
in [13, 33], then by the Functional Central Limit Theorem the limit of this
sequence as $N\to\infty$ 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 $\nu_{p}$ 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 $\nu_{p}$, choose a random vector $v$ in the
tangent space, go the distance $F(v)/\sqrt{N}$ along the geodesic starting at
$p$ with the initial velocity $v$ and then repeat the procedure (if $\nu_{p}$
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 $\mu$, and an extra data
$u_{0}\in H^{1}_{0}(M)$ 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 $\nu_{p}$ 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 $\nu_{p}$ 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 $\nu_{p}$, 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
$\gamma\colon[0,\varepsilon]\rightarrow M$ of a (smooth) Riemannian metric.
Take a vector $v\in T_{\gamma(0)}M$ of length one and its parallel transport
$v_{\varepsilon}\in T_{\gamma(0)}M$ along the geodesic segment. Next, consider
the arc-length parametrised geodesic geodesics $\gamma_{v}$ and
$\gamma_{v_{\varepsilon}}$ which start from $\gamma(0)$ and
$\gamma(\varepsilon)$ with the initial vectors $v$ and $v_{\varepsilon}$,
respectively. Then the distance between $\gamma_{v}(t)$ and
$\gamma_{v_{\varepsilon}}(t)$, behaves, for $\varepsilon\to 0$ and $t\to 0$,
as $\varepsilon(1+Ct^{2})$. In the Euclidean case, the distance does not
depend on $t$ at all and is equal to $\varepsilon$. 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 $\nu_{p}$ on each
$T_{p}M$ is constructed by $F_{|T_{p}M}$ (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].

### Acknowledgements.

We thank M. von Renesse for useful discussions. T.M. and V.M. thank the DFG
for the financial support (Einzelprojekt MA 2565/6).

##  2 Setting and results.

###  2.1 Finsler manifolds.

First we recall the basic definitions in Finsler geometry. Let $M\coloneqq
M^{m}$ be a $m$-dimensional manifold, $m\geq 1$. Suppose that
$(x^{1},\dots,x^{m})$ is a local coordinate at some $p\in M$. Then
$y_{i}=\partial x_{i}$ induces a local coordinate
$(x^{1},\dots,x^{m},y^{1},\dots,y^{m})$ on $TM$. For simplicity, for a
function $H\colon TM\to\mathbb{R}$ we use the notations
$H_{x^{i}}=\partial_{x^{i}}H$ and $H_{y^{i}}=\partial_{y^{i}}H$.

A smooth _Finsler manifold_ $(M,F)$ is a smooth manifold $M$ together with a
continuous function $F\colon TM\to\mathbb{R}_{\geq 0}$ called the _Finsler
metric_ (Finsler function) satisfying the following conditions:

Regularity:

The function $F$ is smooth on $TM\setminus\\{0\\}$.

Positive Homogeneity:

For any $(x,y)\in T_{x}M$ and $\lambda\geq 0$, we have $F(x,\lambda y)=\lambda
F(x,y)$.

Strong Convexity:

For $0\neq(x,y)\in T_{x}M$, the _fundamental tensor_ defined by

$\displaystyle[g_{(x,y)}]_{ij}\coloneqq\left(\dfrac{1}{2}F^{2}\right)_{y^{i}y^{j}}$
(2.1)

is strictly positive definite.

The indicatrix bundle of $(M,F)$ is defined by

$\displaystyle IM=\\{Y\in TM\colon F(Y)=1\\}.$

For any $p\in M$, the fibre $I_{p}M$ of $IM$ is a convex hypersurface in
$T_{p}M$ diffeomorphic to $\mathbb{S}^{m-1}$.

If $(M,\mathbf{g})$ is a Riemannian manifold, one can naturally endow it with
a Finsler metric by setting $F(Y):=\sqrt{\mathbf{g}(Y,Y)}$, $Y\in TM$.
Conversely, a Finsler function corresponds to some Riemannian metric
$\mathbf{g}$ 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 $\gamma\colon[a,b]\rightarrow M$
is a geodesic, if it is a stationary point of the energy functional

$E[\gamma]\coloneqq\dfrac{1}{2}\int_{a}^{b}F^{2}(\gamma(t),\dot{\gamma}(t))\,\mathrm{d}t.$
(2.2)

among all piecewise smooth curves starting at $\gamma(a)$ and ending at
$\gamma(b)$. It is known that for any $p\in M$ and for any $Y\in T_{p}M$,
there exists a unique geodesic $\gamma_{Y}=\gamma_{Y}(t)$ such that
$\gamma(0)=p$ and $\dot{\gamma}(0)=Y$. We define the exponential map at $p$ to
be

$\displaystyle\exp_{p}\colon T_{p}M\ni Y\mapsto\gamma_{Y}(1)\in M$ (2.3)

for all $Y\in T_{p}M$ such that $\gamma_{Y}(t)$ is defined for $t\in[0,1]$. We
say $(M,F)$ is forward complete if for any $p\in M$ the exponential map
$\exp_{p}$ is defined for all $Y\in T_{p}M$. The manifold $(M,F)$ is
geodesically complete, if each geodesic $\gamma$ can be extended to a geodesic
defined for all $t\in(-\infty,\infty)$.

For a piecewise smooth curve $\gamma\colon[a,b]\rightarrow M$, its length is
defined by

$\displaystyle\mathbf{Length}(\gamma)=\int_{a}^{b}F(\gamma(t),\dot{\gamma}(t))\,\mathrm{d}t.$
(2.4)

The Finsler function $F$ defines the following asymmetric and symmetrized
distances on $M$:

$\displaystyle d_{a}(p,q)$
$\displaystyle\coloneqq\inf\Big{\\{}\textbf{Length}(\gamma)\colon\gamma\text{
is a piecewise smooth curve from $p$ to $q$}\Big{\\}},$ $\displaystyle d(p,q)$
$\displaystyle\coloneqq\max\\{d_{a}(p,q),d_{a}(q,p)\\}$ (2.5)

By the Hopf–Rinow theorem for Finsler manifolds (see e.g. [5, Section 6.6]),
if $(M,F)$ is forward complete, the metric space $(M,d)$ is complete. For a
forward complete $(M,F)$, every closed ball of $(M,d)$ is compact. The
manifold $M$ can be naturally endowed with the Borel sigma-algebra that makes
it a measure space.

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. $F(x,y)\not\equiv F(x,-y)$, 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.

Hc: The manifold $(M,F)$ is connected and forward complete.

Hb: 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. 1.

Uniform ellipticity: There is some constant $C>1$ such that for any $p\in M$
and any non-zero $u,v\in T_{p}M$, we have

$\displaystyle\frac{1}{C^{2}}F^{2}(v)=\dfrac{1}{C^{2}}g_{v}(v,v)\leq
g_{u}(v,v)\leq C^{2}g_{v}(v,v)=C^{2}F^{2}(v).$ (2.6)

  2. 2.

The flag curvature $K$ is bounded uniformly and absolutely by some constant
$\lambda>0$, namely $\lVert K\rVert\leq\lambda$.

  3. 3.

The $T$-curvature is also bounded uniformly and absolutely in the following
sense. For any $p\in M$, any $u,v\in T_{p}M$ with $F(v)=1$, the $T$-curvature
satisfies

$\displaystyle\lvert
T_{v}(u)\rvert\leq\lambda\\{g_{v}(u,u)-[g_{v}(u,v)]^{2}\\}$ (2.7)

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 $f\colon M\rightarrow\mathbb{R}$ vanishes at infinity, if
$\forall\varepsilon>0$, there exists some compact set $K_{\varepsilon}\subset
M$ such that $\|f\|\leq\varepsilon$ outside $K_{\varepsilon}$.

We denote the unit discs on $TM$ by

$\displaystyle D_{p}M=\\{Y\in T_{p}M\colon F(Y)\leq 1\\}.$

Let $d$ be the symmetrized distance defined by (2.5). We denote the open balls
by:

$\displaystyle B_{p}(\varepsilon)=\\{q\in M\colon
d(p,q)<\varepsilon\\},\quad\varepsilon>0.$

In this paper $\mathcal{B}$ is the space of Borel measurable real valued
functions on $M$, $\mathcal{C}_{0}$ is the space of continuous that vanish at
infinity, $\mathcal{C}^{\infty}$ is the space of smooth functions,
$\mathcal{C}^{\infty}_{K}$ is the space of smooth functions with compact
support. Furthermore, $D([0,\infty),M)$ is the collection of right continuous
functions $\gamma\colon[0,\infty)\rightarrow M$ with left limits, and
$C([0,\infty),M)$ is space of continuous functions
$\gamma\colon[0,\infty)\rightarrow M$.

###  2.2 Rescaled geodesic random walks.

As a motivation for “rescaling”, let us consider the following example of a
geodesic random walk: The manifold is $\mathbb{R}$ with the standard flat
metric and $\nu_{p}$ is defined as follows. At every point $p$, the support of
$\nu_{p}$ is two vectors $\\{1,-1\\}\subset T_{p}\mathbb{R}=\mathbb{R}$ such
that the probability of $-1$ is $1/4$, and probability of $1$ is $3/4$. That
is, the particle goes the distance $1/\sqrt{N}$ 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
$1/(2\sqrt{N})$ in the positive direction, so in $N$ steps the mean value is
$\sqrt{N}/2$. For $N\to\infty$, most trajectories “escape to infinity”, so the
limit of the sequence of such stochastic processes for $N\to\infty$ does not
exist.

This phenomenon appears in all dimensions when the probability distribution
$\nu_{p}$ (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 $\nu_{p}$ by
$\mu_{p}$, and modify the measure $\nu_{p}$ by shifting it in $T_{p}M$ by the
vector $-\mu_{p}+\mu_{p}/\sqrt{N}$, i.e.
$\tilde{\nu}_{p}(y):=\nu_{p}(y-\mu_{p}+\mu_{p}/\sqrt{N})$. 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 $-\mu_{p}$. 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 $\mu_{p}$ are
centrally-symmetric 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 $\nu_{p}$. Indeed, for $N=1$ we have
$-\mu_{p}+\mu_{p}/\sqrt{N}=0$. We also feel that in the case that Finsler
geodesic random walk is used to describe a physical model, the mean of
$\nu_{p}$ should somehow come in the definition. Of course, it may come with
any other coefficient $\alpha$, i.e., $\nu_{p}$ may be shifted by the vector
$-\mu_{p}+\alpha\mu_{p}/\sqrt{N}$. But also in this case (even if $\alpha$
depends on the position) our results are applicable. Indeed, if we modify
$\nu_{p}$ by shifting it by $\beta\mu_{p}$, then the rescaled measure will be
shifted by $(1+\beta)\mu_{p}/\sqrt{N}$. Thus, all results of our paper can be
applied for any $\alpha$, in particular for $\alpha=0$.

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.

###  2.3 The main result.

We will assume that the Finsler manifold $(M,F)$ is connected and forward
complete (Hypothesis Hc) and has bounded geometry (Hypothesis Hb). In
addition, we make the following assumption on the family of measures
$\\{\nu_{p}\\}$.

Hν: We assume that $\nu=\\{\nu_{p}\\}$ is a smooth family of probability
measures inside $DM:=\\{Y\in TM\mid F(Y)\leq 1\\}$ in $TM$ or on the
$F$-indicatrices.

In the first case we require that $\nu$ is a smooth $m$-form on $DM$ such that
for every $p$, the restriction $\nu_{p}:=\nu|_{T_{p}M}$ is a form on the disc
$D_{p}M:=\\{Y\in T_{p}M\mid F(Y)\leq 1\\}$ inducing a probability measure.
Similarly, in the second case $\nu$ is a smooth $(m-1)$-form on $IM$ such that
for each $p\in M$ the restriction $\nu_{p}:=\nu|_{I_{p}M}$ 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:

######  Theorem 2.1.

Let Hypotheses $\mathbf{H}_{c}$, $\mathbf{H}_{b}$ and $\mathbf{H}_{\nu}$ be
satisfied. Consider a family of geodesic random walks starting at $p_{0}$
constructed from $(M,F)$ and $\\{\nu_{p}\\}_{p\in M}$. Then, this sequence has
a unique weak limit $\xi$. The process $\xi$ is a diffusion whose generator is
a non-degenerate elliptic differential operator $A$ with smooth coefficients
given by

$\displaystyle
Af(p)=\mathrm{d}f(\mu_{p})+\dfrac{1}{2}\int_{T_{p}M}\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}f\circ\gamma_{Y-\mu_{p}}(t)\nu_{p}(\mathrm{d}Y),\quad
f\in\mathcal{C}^{\infty}_{K}.$ (2.8)

Here $\gamma_{Y-\mu_{p}}$ is the geodesic with initial vector $Y-\mu_{p}$. In
the local coordinates, it has the following form:

$\displaystyle Af(p)=$ $\displaystyle
f_{k}\left(\mu_{p}^{k}-\dfrac{1}{2}\int_{T_{p}M}\Gamma^{k}_{ij}\left(p,y-\mu_{p}\right)(y^{i}-\mu_{p}^{i})(y^{j}-\mu_{p}^{j})\,\nu_{p}(\mathrm{d}y)\right)$
(2.9)
$\displaystyle+\dfrac{1}{2}f_{ij}\int_{T_{p}M}(y^{i}-\mu_{p}^{i})(y^{j}-\mu_{p}^{j})\,\nu_{p}(\mathrm{d}y),$

where $\Gamma^{k}_{ij}$ are the formal Christoffel symbols of the second kind
given by

$\displaystyle\Gamma^{k}_{ij}(x,y)=\dfrac{1}{2}g^{ks}\left(\dfrac{\partial
g_{is}}{\partial x_{j}}+\dfrac{\partial g_{js}}{\partial
x_{i}}-\dfrac{\partial g_{ij}}{\partial x_{s}}\right)(x,y),\quad y\neq 0,$

$f_{k}=\partial_{x^{k}}f$ and $f_{ij}=\partial^{2}_{x^{i}x^{j}}f$.

Moreover, $\xi$ is stochastically complete.

####  2.3.1 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 $\nu_{p}$ is invariant with respect to the parallel transport along
any curve and is invariant with respect to the standard action of
$SO(\mathbf{g})$ on $T_{p}M$. Actually, for a generic metric invariance with
respect to the parallel transport implies $SO(\mathbf{g})$-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 $\sigma(A)$ of $A$ is dual to a
Riemannian metric on $M$ which we denote $\mathbf{g}_{A}$. Then the
Beltrami–Laplace operator $\Delta^{A}$ of $\mathbf{g}_{A}$ and $A$ have the
same symbol. Hence $A-\Delta^{A}$ 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 $\nu_{p}$ 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.

####  2.3.2 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 $SO(\mathbf{g})$, which implies that in the
orthogonal coordinates $(y^{1},\dots,y^{m})$ on $T_{p}M$ it is given by
$\phi((y^{1})^{2}+\cdots+(y^{m})^{2})\,\mathrm{d}y^{1}\wedge\cdots\wedge\mathrm{d}y^{m}$.
The function $\phi$ is the same for all points $p$, has the property that it
is nonnegative and that the integral
$\int_{\mathbb{R}^{n}}\phi((y^{1})^{2}+\cdots+(y^{m})^{2})\,\mathrm{d}y^{1}\wedge\cdots\wedge\mathrm{d}y^{m}=1$.

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 $p\in M$, let
$\omega^{\prime}_{p}=\mathrm{d}y^{1}\wedge\cdots\wedge\mathrm{d}y^{m}$ 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 $U\subset T_{p}M$ it the Lebesgue-measure of the intersection
$D_{p}M\cap U$), and normalize it such that it becomes a probability measure.

Measure coming from the fundamental tensor:

For any $p\in M$, 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

$\displaystyle\nu_{p}\coloneqq\dfrac{\mathrm{vol}_{g}}{\mathrm{vol}_{g}(I_{p}M)}.$
(2.10)

This probability measure is close to the one used in [4, §A2].

Measure coming from the Hilbert form:

Denote $\mathbb{P}^{+}(M)$ the positive projectivized tangent bundle. The
Hilbert 1-form $\hat{\omega}=F_{y^{i}}\,\mathrm{d}x^{i}$ defined on
$TM\setminus\\{0\\}$ is actually a pull back of some 1-form $\omega$ on
$\mathbb{P}^{+}(M)$ by the standard projection. It is well known that
$\omega\wedge(\mathrm{d}\omega)^{m-1}$ defines a volume form on
$\mathbb{P}^{+}(M)\simeq IM$. Let $\mathrm{i}_{p}\colon I_{p}M\rightarrow IM$
be the standard inclusion and $\pi\colon IM\rightarrow M$ be the canonical
projection. It is known (see e.g. [7]) that there exists a $(m-1)$-form
$\alpha^{F}$ on $IM$ and a volume form $\omega^{F}$ on $M$ such that
$\alpha^{F}|_{I_{p}M}$ is a unique volume form on $I_{p}M$ for each $p\in M$
with

$\displaystyle\mathrm{vol}_{\alpha^{F}}(I_{p}M)=1,$ (2.11)
$\displaystyle\alpha^{F}\wedge\pi^{*}\omega^{F}=\omega\wedge(\mathrm{d}\omega)^{m-1}.$

Hence we can take $\nu_{p}\coloneqq\mathrm{vol}_{\alpha^{F}}$ on $I_{p}M$.

Each of these measures satisfies the Hypothesis $\mathbf{H}_{\nu}$, 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 $\nu_{p}$ may
also be useful in Finsler geometry.

###  2.4 Example: Limit diffusion for a Katok Finsler metric.

Let $(\mathbb{S}^{2},\mathbf{g})$ 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 $\mathbf{g}(X,X)<1$. In the the following spherical
coordinate on $\mathbb{S}^{2}$,

$\displaystyle(\psi,\theta)\mapsto(\cos(\psi)\cos(\theta),\sin(\psi)\cos(\theta),\sin(\theta)).$
(2.12)

We have

$\displaystyle X=r\partial_{\psi},\ \|r\|<1.$

Now for any $p\in M$, the indicatrix $I_{p}M$ of the constructed Finsler
function $F$ is obtained by shifting the unit sphere $S_{p}\mathbb{S}^{2}$ of
$\mathbf{g}$ (which is the indicatrix with respect to $\mathbf{g}$) by $X$.
That is,

$\displaystyle I_{p}M\coloneqq\\{v+X_{p}\colon v\in T_{p}\mathbb{S}^{2},\
\mathbf{g}(v,v)=1\\}$

This yields a well-defined Finsler metric as $\mathbf{g}(X,X)<1$. 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 $\nu_{p}$ we consider the Lebesgue measure as described in
§2.3.2; let us calculate the generator of the corresponding diffusion process
$\xi$.

By Theorem 2.1, the diffusion process $\xi$ generated by $\\{\nu_{p}\\}_{p\in
M}$ has generator $A$ such that

$\displaystyle Af(p)=$
$\displaystyle\mathrm{d}f(X)(p)+\dfrac{1}{2}\biggr{\\{}f_{ij}\int_{D_{p}M}(Y^{i}-X^{i})(Y^{j}-X^{j})\,\nu_{p}(\mathrm{d}Y)$
(2.13) $\displaystyle-
f_{k}\int_{D_{p}M}\Gamma^{k}_{ij}\left(p,Y-X\right)(Y^{i}-X^{i})(Y^{j}-X^{j})\,\nu_{p}(\mathrm{d}Y)\biggr{\\}}$

Here $\Gamma^{k}_{ij}$ are the formal Christoffel symbols of the second kind
of $(M,F)$ and $f\in\mathcal{C}^{\infty}$.

As $\nu_{p}$ is induced by a Lebesgue measure on $T_{p}M\simeq\mathbb{R}^{m}$,
we also denote this Lebesgue measure by $\nu_{p}$ for simplicity. For any
$p\in M$, the set

$\displaystyle\hat{D}_{P}M\coloneqq\\{Y\in T_{p}M\ |\ Y+X(p)\in D_{p}M\\}$

is just the closed unit ball on $T_{p}\mathbb{S}^{2}$ with respect to
$\mathbf{g}$. Since $\nu_{p}$ is translation invariant, Equation (2.13)
becomes

$\displaystyle Af(p)=$
$\displaystyle\mathrm{d}f(X)(p)+\dfrac{1}{2}\biggr{\\{}f_{ij}\int_{\hat{D}_{p}M}Y^{i}Y^{j}\nu_{p}(\mathrm{d}Y)-f_{k}\int_{\hat{D}_{p}M}\Gamma^{k}_{ij}\left(p,Y\right)Y^{i}Y^{j}\,\nu_{p}(\mathrm{d}Y)\biggr{\\}}$
(2.14)

Note the integrand in the equation above is second order homogeneous in $Y$.
By Fubini theorem, for any $p\in M$ there is a finite measure $\eta_{p}$ on
$S_{p}\mathbb{S}^{2}$ such that for any integrable second order homogeneous
function $h$ on $T_{p}M$, we have

$\displaystyle\int_{\hat{D}_{p}M}h(Y)\nu_{p}(\mathrm{d}Y)=\int_{S_{p}\mathbb{S}^{2}}h(Y)\eta_{p}(\mathrm{d}Y)$
(2.15)

Because $\nu_{p}$ is invariant under any orthogonal transformation on
$T_{p}\mathbb{S}^{2}$ with respect to $\mathbf{g}$, it is clear $\eta_{p}$ is
a multiple of the canonical angular measure $m_{p}$ on $S_{p}\mathbb{S}^{2}$
with respect to $\mathbf{g}$. A straight forward computation shows
$\eta_{p}=\dfrac{1}{4\pi}m_{p}$. Hence from (2.14), we get

$\displaystyle
Af(p)=\mathrm{d}f(X)(p)+\dfrac{1}{8\pi}\biggr{\\{}f_{ij}\int_{S_{p}\mathbb{S}^{2}}Y^{i}Y^{j}m_{p}(\mathrm{d}Y)-f_{k}\int_{S_{p}\mathbb{S}^{2}}\Gamma^{k}_{ij}\left(p,Y\right)Y^{i}Y^{j}\,m_{p}(\mathrm{d}Y)\biggr{\\}}$
(2.16)

Let $\Delta$ be the Beltrami–Laplace operator of $\mathbf{g}$, and let
$\hat{\Gamma}^{k}_{ij}$ be the Christoffel symbols of $\mathbf{g}$. A
straightforward computation yields

$\displaystyle\dfrac{1}{8}\Delta
f(p)=\dfrac{1}{8\pi}\biggr{\\{}f_{ij}\int_{S_{p}\mathbb{S}^{2}}Y^{i}Y^{j}\,m_{p}(\mathrm{d}Y)-f_{k}\int_{S_{p}\mathbb{S}^{2}}\hat{\Gamma}^{k}_{ij}(p)Y^{i}Y^{j}\,m_{p}(\mathrm{d}Y)\biggr{\\}}.$
(2.17)

This implies $\dfrac{1}{8}\Delta$ and $A$ have the same symbol. To compute the
drift of $A$, we assume without loss of generality that $0\leq r<1$. First we
have

$\displaystyle\left(A-\dfrac{1}{8}\Delta\right)f(p)$
$\displaystyle=\mathrm{d}f(X)(p)+\dfrac{1}{8\pi}f_{k}\int_{S_{p}\mathbb{S}^{2}}\hat{\Gamma}^{k}_{ij}(p)Y^{i}Y^{j}\,m_{p}(\mathrm{d}Y)$
(2.18)
$\displaystyle-\dfrac{1}{8\pi}f_{k}\int_{S_{p}\mathbb{S}^{2}}\Gamma^{k}_{ij}\left(p,Y\right)Y^{i}Y^{j}\,m_{p}(\mathrm{d}Y).$

Let $\Phi_{t}$ be the flow generated by $X$, we know from [16, Theorem 1] that
if $\gamma(t)$ is a geodesic of $(\mathbb{S}^{2},\mathbf{g})$ with
$\sqrt{\mathbf{g}(\dot{\gamma},\dot{\gamma})}=c$, then
$\hat{\gamma}(t)=\Phi_{ct}\circ\gamma(t)$ is a geodesic of $F$ with initial
vector $\hat{\gamma}^{\prime}(0)=\dot{\gamma}(0)+cX(\gamma(0))$. But in the
spherical coordinate given (2.12), the flow $\Phi_{t}$ simply has the form

$\displaystyle\Phi_{t}(\psi,\theta)=(\psi+rt,\theta).$

Then in this coordinate, we have

$\displaystyle\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}\hat{\gamma}(t)=\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}\gamma(t).$

By the geodesic equation, this is equivalent to

$\displaystyle\Gamma^{k}_{ij}(p,\hat{\gamma}^{\prime}(0))(\hat{\gamma}^{\prime}(0))^{i}(\hat{\gamma}^{\prime}(0))^{j}=\hat{\Gamma}^{k}_{ij}(p)(\gamma^{\prime}(0))^{i}(\gamma^{\prime}(0))^{j}.$

Using this and (2.18), a straight forward computation gives

$\displaystyle\Big{(}A-\frac{1}{8}\Delta\Big{)}=X+\dfrac{1}{4}r^{2}\cos(\theta)\sin(\theta)\cdot\dfrac{r^{2}\cos^{2}(\theta)-2}{\left(1-r^{2}\cos^{2}(\theta)\right)^{2}}\cdot\partial_{\theta}.$
(2.19)

This is the drift of the generator $A$.

Figure 2:  (l.) A sample path of the Brownian motion on the standard sphere.
(r.) A sample path of the diffusion with the generator $A$ given by (2.13)
with $r=1/2$.

On Figure 2 one clearly sees the difference in the behaviours of the Brownian
motion of the initial round metric on $\mathbb{S}^{2}$ 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.

##  3 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.

###  3.1 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:

$\displaystyle\frac{\mathrm{d}x^{i}}{\mathrm{d}t}=y^{i},$ (3.1)
$\displaystyle\frac{\mathrm{d}y^{k}}{\mathrm{d}t}+\Gamma^{k}_{ij}(x,y)y^{i}y^{j}=0.$
(3.2)

Here $\Gamma^{k}_{ij}$ are the formal Christoffel symbols of the 2nd kind:

$\displaystyle\Gamma^{k}_{ij}(x,y)=\dfrac{1}{2}g^{ks}\left(\dfrac{\partial
g_{is}}{\partial x_{j}}+\dfrac{\partial g_{js}}{\partial
x_{i}}-\dfrac{\partial g_{ij}}{\partial x_{s}}\right)(x,y),\quad y\neq 0$
(3.3)

It is immediate from (3.3) that for any $\lambda>0$ and $y\neq 0$, we have

$\displaystyle\Gamma^{k}_{ij}(x,y)=\Gamma^{k}_{ij}(x,\lambda y).$ (3.4)

Denote the class of real-valued $k$-times continuously differentiable real-
valued functions on $M$ with compact support by $\mathcal{C}^{k}_{K}$. Suppose
$f\in\mathcal{C}^{k_{0}}_{K}$ with compact support $K_{f}$. For any $Y\in TM$,
define $f_{Y}(t)=f\circ\gamma_{Y}(t)$, where $\gamma_{Y}(t)$ is the geodesic
with initial vector $Y$ as before.

######  Lemma 3.1.

Suppose that $f\in\mathcal{C}^{k_{0}}_{K}$. There exists some constant $c$
such that

$\displaystyle\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(t)\biggr{\rvert}\leq
cF^{k}(Y),\quad\forall k\leq k_{0},$ (3.5)

wherever it is well defined.

###### Proof.

First we show there is some constant $c$ such that
$\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(0)\biggr{\rvert}\leq
cF^{k}(Y)$. Let $\tilde{X}$ be the geodesic spray on $TM$, and denote
$\pi\colon TM\rightarrow M$ the canonical projection. The function $\pi^{*}f$
is $\mathcal{C}^{k_{0}}$, so $(\mathcal{L})^{k}_{\tilde{X}}(\pi^{*}f)$ is
continuous on $TM\setminus\\{0\\}$ for any $k\leq k_{0}$. Hence for some
constant $c$, we have

$\lvert(\mathcal{L})^{k}_{\tilde{X}}(\pi^{*}f)(\hat{p})\rvert\leq c,\
\forall\hat{p}\in IK_{f},\ \forall k\leq k_{0}.$

For any $p\in M$ and $Y\in T_{p}M$, we have

$\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(0)=(\mathcal{L})^{k}_{\tilde{X}}(\pi^{*}f)(p,Y).$

In addition, for $Y\neq 0$, let $Y^{\prime}=\dfrac{Y}{F(Y)}$. Using

$f_{Y}(t)=f\circ\gamma_{F(Y)Y^{\prime}}(t)=f_{Y^{\prime}}(F(Y)t),$

we get

$\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(0)\biggr{\rvert}=\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y^{\prime}}\right)(0)\biggr{\rvert}F^{k}(Y).$

Then for $p\in K_{f}$ and $Y\neq 0$, we have

$\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(0)\biggr{\rvert}=(\mathcal{L})^{k}_{\tilde{X}}(\pi^{*}f)(p,Y^{\prime})\cdot
F^{k}(Y)\leq cF^{k}(Y).$

For $p\notin K_{f}$, we have $f$ vanishes identically on some neighbourhood of
$p$. Then the function $t\mapsto f_{Y}(t)$ is constant near $t=0$. For $Y=0$,
then function $f_{Y}(t)$ is also constant. It follows that
$\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(0)\biggr{\rvert}\leq
cF^{k}(Y)$.

Next, given any geodesic $\gamma_{Y}$, let $Y(t)$ be its velocity field. We
have

$F(Y(t))=F(Y(0))=F(Y),$
$\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y}\right)(t)\biggr{\rvert}=\biggr{\lvert}\left(\dfrac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}f_{Y(t)}\right)(0)\biggr{\rvert}\leq
cF^{k}(Y(t))=cF^{k}(Y).$

This completes the proof. ∎

The injective radius at $p$ is defined by

$\displaystyle\mathrm{inj}_{M}(p)$
$\displaystyle\coloneqq\inf\\{r>0\colon\exp_{p}|_{D_{p}(r)}\ \mathrm{is\
injective}\\}$ $\displaystyle\mathrm{inj}_{M}$
$\displaystyle\coloneqq\inf\\{\mathrm{inj}_{M}(p)\colon p\in M\\}$

The conjugate radius is defined similarly by

$\displaystyle\mathrm{con}_{M}(p)$
$\displaystyle\coloneqq\inf\\{r>0\colon\exp_{p}|(D_{p}(r))\ \mathrm{is\ an\
immersion}\\}$ $\displaystyle\mathrm{con}_{M}$
$\displaystyle\coloneqq\inf\\{\mathrm{con}_{M}(p)\colon p\in M\\}.$

Clearly we always have $\mathrm{inj}_{M}(p)\leq\mathrm{con}_{M}(p)$ for any
$p\in M$. The conjugate radius and flag curvature 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 flag curvature
$\|K\|\leq\lambda$. Then the conjugate radius is bounded from below by
$\mathrm{con}_{M}\geq\dfrac{\pi}{\sqrt{\lambda}}$, hence strictly positive.

###  3.2 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.

######  Definition 3.1.

Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space. An $M$-valued
process $\xi\colon\Omega\times[0,\infty)\rightarrow M$ is Markovian if for
each Borel subset of $B$ of $M$, for all $n\geq 1$, $0\leq
s_{1}<\cdots<s_{n}<s<t$ we have,

$\displaystyle\mathbf{P}(\xi_{t}\in
B|\xi_{s_{1}},\dots,\xi_{s_{n}},\xi_{s})=\mathbf{P}(\xi_{t}\in B|\xi_{s}).$
(3.6)

The transition probability function for a Markov process is defined by

$\displaystyle P(p,s,t,B)=\mathbf{P}(\xi_{t}\in B|\xi_{s}=p),\ \forall p\in
M,\ \forall 0\leq s\leq t.$

We say $(\xi_{t})_{t\geq 0}$ is time homogeneous if the following holds.

$\displaystyle P(p,s,t,B)=P(p,0,t-s,B),\ \forall p\in M,\ \forall 0\leq s\leq
t.$

All Markov processes considered in this paper are time-homogeneous. A time
homogenous Markov process $\xi$ defines a semigroup $T=(T_{t})_{t\geq 0}$ of
linear operators on the measurable functions $\mathcal{B}$ on $M$ by

$\displaystyle T_{t}(f)(p)=\mathbf{E}_{p}[f(\xi_{t})],\ p\in M,\ t\geq 0.$

We say a Markov process $\xi$ is Feller if the semigroup $T$ is a strongly
continuous semigroup of positive contractions on the Banach space
$\mathcal{C}_{0}$.

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
$\\{\nu_{p}\\}_{p\in M}$ be a family of measures such that each $\nu_{p}$ is a
probability measure on $T_{p}M$. Denote by $\mu_{p}$ the mean of $\nu_{p}$

$\displaystyle\mu_{p}\coloneqq\int_{T_{p}M}Y\,\nu_{p}(\mathrm{d}Y).$

In our setting (Hypothesis $\mathbf{H}_{\nu}$), the probability measures are
compactly supported so $\mu_{p}$ exists and is finite.

######  Definition 3.2.

Let $N\geq 1$ and let $p_{0}\in M$ be fixed. A random process
$(\zeta_{k}^{N},Y_{k+1}^{N})_{k\geq 0}$ is called a (rescaled) discrete time
geodesic random walk on $M$ with initial point $p_{0}$ and with increments
$\\{Y_{k+1}^{N}\\}_{k\geq 0}$ compatible with the family $\\{\nu_{p}\\}_{p\in
M}$ if

  1. 1.

the process $\zeta_{k}^{N}$ is $M$-valued, and $Y_{k+1}^{N}$ is
$T_{\xi_{k}^{N}}M$-valued,

  2. 2.

$\zeta_{0}^{N}=p_{0}$,

  3. 3.

for each $k\geq 0$, $\text{Law}(Y_{k+1}^{N})=\nu_{\zeta_{k}^{N}}^{N}$ where

$\displaystyle\nu_{p}^{N}(B)$
$\displaystyle=\int_{T_{p}M}\mathbb{I}_{B}\Big{(}\frac{Y-\mu_{p}}{\sqrt{N}}+\frac{\mu_{p}}{N}\Big{)}\,\nu_{p}(\mathrm{d}Y)$
(3.7)

for any measurable $B\subseteq T_{p}M$,

  4. 4.

$\zeta_{k+1}^{N}=\exp_{\zeta_{k}^{N}}(Y_{k+1}^{N})$, $k\geq 0$.

Hence the processes $\zeta^{N}$ are defined by the family of measures
$\\{\nu_{p}\\}_{p\in M}$ and the geometry of the exponential mapping
$\exp_{p}$. The random walks are time-homogeneous since the family
$\\{\nu_{p}\\}_{p\in M}$ 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 $Y_{k+1}^{N}$ depend only on the
current position $\zeta^{N}_{k}$ and not on the previous positions and
increments. More precisely, we introduce the natural filtration

$\displaystyle\mathcal{F}_{k}^{N}:=\sigma\\{(\zeta_{0}^{N},Y^{N}_{1}),\dots,(\zeta^{N}_{k-1},Y_{k}^{N})\\},\quad
k\geq 1,$ (3.8)

and say that the increments of $\zeta^{N}$ are independent if for each
$f\in\mathcal{C}_{b}(\oplus_{i=0}^{k+1}M,\mathbb{R})$

$\displaystyle\mathbf{E}\Big{[}f(\zeta_{0}^{N},\dots,\zeta_{k}^{N},\zeta_{k+1}^{N})\Big{|}\mathcal{F}_{k}^{N}\Big{]}=\int_{T_{\zeta_{k}^{N}}M}f(\zeta_{0}^{N},\dots,\zeta_{k}^{N},\exp_{\zeta_{k}^{N}}(Y))\,\nu_{\zeta_{k}^{N}}^{N}(\mathrm{d}Y).$
(3.9)

It is clear that $\zeta^{N}$ is a homogeneous discrete time $M$-valued Markov
chain with the one-step transition operator

$\displaystyle
P^{N}f(p)=\mathbf{E}_{p}f(\zeta^{N}_{1})=\int_{T_{p}M}f\Big{(}\exp_{p}\Big{(}\frac{Y-\mu_{p}}{\sqrt{N}}+\frac{\mu_{p}}{N}\Big{)}\Big{)}\,\nu_{p}(\mathrm{d}Y),\quad
f\in\mathcal{C}_{b}(M,\mathbb{R}).$ (3.10)

Since we work in a continuous time setting it is convenient to transform the
discrete time Markov chain $\zeta^{N}$ into a continuous time Markov process.
This can be done by a standard subordination procedure.

Let $Q=(Q_{t})_{t\geq 0}$ be a standard Poisson process independent of
$\\{\zeta^{N}\\}$. Define a _pseudo-Poisson_ process

$\displaystyle\xi^{N}_{t}=\zeta^{N}_{Q_{Nt}},\quad t\geq 0.$

Note that the sample paths of $\xi^{N}$ belong to $D([0,\infty),M)$. Hence,
the Markov processes $\xi^{N}$ induce probability distributions
$\mathbf{P}^{N}$ on the path space $D([0,\infty),M)$. It is easy to see that
the transition semigroup $T^{N}=(T^{N}_{t})_{t\geq 0}$ of $\xi^{N}_{t}$ has
the form

$\displaystyle
T^{N}_{t}(f)(p)=\mathbf{E}_{p}[f(\xi^{N}_{t})]=\mathrm{e}^{-Nt}\sum_{k=0}^{\infty}\dfrac{(Nt)^{k}}{k!}(P^{N})^{k}(f)(p),\
f\in\mathcal{B}.$ (3.11)

Finally we introduce a of family continuous $M$-valued processes defined by

$\displaystyle\hat{\xi}_{t}^{N}$
$\displaystyle=\exp_{\zeta^{N}_{k}}\Big{(}N\Big{(}t-\frac{k}{N}\Big{)}Y^{N}_{k+1}\Big{)},\quad
t\in\Big{[}\frac{k}{N},\frac{k+1}{N}\Big{]},\quad k\geq 0.$

Since the manifold is geodesically complete, the processes $\zeta^{N}$,
$\xi^{N}$ and $\hat{\xi}^{N}$ are well-defined for each $N\geq 1$. By
construction the processes $\hat{\xi}^{N}$ have piecewise smooth sample paths
consisting of geodesic segments, and induce probability distributions
$\hat{\mathbf{P}}^{N}$ on the path space $C([0,\infty),M)$ 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 processes $\hat{\xi}^{N}$ are not Markovian, the convergence of
the continuous time processes $(\zeta^{N}_{[Nt]})_{t\geq 0}$,
$(\xi^{N}_{t})_{t\geq 0}$ and $(\hat{\xi}^{N}_{t})_{t\geq 0}$ is equivalent,
see, e.g. [23, Theorem 17.28]. In the next sections we will mainly work with
the Markov processes $\xi^{N}$.

##  4 Proof of the main theorem.

In this section we prove the convergence of the geodesic random walks
$\\{\xi^{N}\\}$. We will assume that the Finsler manifold $(M,F)$ is forward
complete and connected (Hypothesis Hc) and has bounded geometry (Hypothesis
Hb) and that the measures $\\{\nu_{p}\\}_{p\in M}$ satisfy the condition Hν
from §2.3.

###  4.1 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.

######  Lemma 4.1.

Let $\mathbf{H}_{\nu}$ hold true and $k\geq 0$. For any
$\mathcal{C}^{k}$-smooth function $f\colon TM\to\mathbb{R}$ the mapping

$\displaystyle p\to\int_{T_{p}M}f(Y)\,\nu_{p}(\mathrm{d}Y)$ (4.1)

is also $\mathcal{C}^{k}$-smooth.

This lemma is obvious since each $\nu_{p}$ 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.

######  Proposition 4.2.

Suppose $(M,F)$ is complete and uniform elliptic. In addition, assume the
measures $\\{\nu_{p}\\}$ satisfy the hypothesis $\mathbf{H}_{\nu}$. Then for
each $N\geq 1$, the family of operators $T^{N}=(T^{N}_{t})_{t\geq 0}$ is a
conservative Feller semigroup with the generator

$\displaystyle A_{N}f=N\Big{(}P^{N}f-f\Big{)},\quad f\in\mathcal{C}_{0}.$
(4.2)

###### Proof.

Let $N\geq 1$ 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 $\mathbf{H}_{c}$.
Let us show that $T^{N}_{t}$ maps $\mathcal{\mathcal{missing}}{C}_{0}$ into
itself for $t\geq 0$. Since we have

$\displaystyle\|P^{N}f\|\leq\|f\|,$ (4.3)

the series in (3.11) converges uniformly. It suffices to show $P^{N}$ maps
$\mathcal{C}_{0}$ into itself.

By Lemma 4.1 the mean value $\mu_{p}$ is a $\mathcal{C}^{\infty}$ vector
field. Since the exponential map for Finsler manifold is at least
$\mathcal{C}^{1}$, then $P^{N}$ maps continuous functions into continuous
functions.

For any $\varepsilon>0$, choose some compact $K\subseteq M$ such that
$|f(x)|<\dfrac{\varepsilon}{2}$ for $x\notin K$. Fix any $p_{0}\in K$, and
define the closed forward balls at $p_{0}$ for $R\geq 0$ by

$\displaystyle B^{+}_{p_{0}}(R)\coloneqq\\{q\in M\colon d_{a}(p_{0},q)\leq
R\\}.$

Because $K$ is compact, there exists some $R_{0}>0$ such that $K\subset
B^{+}_{p_{0}}(R)$ for all $R\geq R_{0}$. 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 $0\neq Y\in T_{p}M$ and $p\in M$, the uniform ellipticity condition in
Definition 2.1 gives

$\displaystyle F^{2}(-Y)=g_{-Y}(Y,Y)\leq C^{2}g_{Y}(Y,Y)=C^{2}F^{2}(Y).$ (4.4)

It follows that

$\displaystyle d_{a}(q,p_{0})$ $\displaystyle\leq Cd_{a}(p_{0},q)\leq CR_{0},\
\forall q\in K;$ (4.5) $\displaystyle d_{a}(p,p_{0})$
$\displaystyle\geq\dfrac{1}{C}d_{a}(p_{0},p)\geq\dfrac{R}{C},\ \forall
p\in(B^{+}_{p_{0}}(R))^{c},\ \forall R\geq 0.$ (4.6)

Let $R_{1}\coloneqq C(C+2+CR_{0})$, then $\forall
p\in(B^{+}_{p_{0}}(R_{1}))^{c}$ and $\forall q\in K$, we have

$\displaystyle d_{a}(p,q)\geq
d_{a}(p,p_{0})-d_{a}(q,p_{0})\geq\dfrac{R_{1}}{C}-CR_{0}>C+1$ (4.7)

On the other hand, for $p\in M$ and $Y\in D_{p}M$, we have

$\displaystyle d_{a}(p,\textbf{e}_{p}^{N}(Y))\leq
F(\dfrac{1}{\sqrt{N}}(Y-(1-1/\sqrt{N})\mu_{p}))\leq F(Y)+F(-\mu_{p})\leq C+1.$
(4.8)

Hence, $\forall Y\in D_{p}M$ and $p\in(B^{+}_{p}(R_{1}))^{c}$, we have
$\textbf{e}_{p}^{N}(Y)\notin K$. It follows that $\forall
p\in(B^{+}_{p}(R_{1}))^{c}$:

$\displaystyle|P^{N}f(p)|=\biggr{\lvert}\int_{T_{p}M}f\circ\textbf{e}_{p}^{N}(Y)\,\nu_{p}(\mathrm{d}Y)\biggr{\rvert}\leq\dfrac{\varepsilon}{2}.$
(4.9)

That is to say $\lVert P^{N}f\rVert\leq\dfrac{\varepsilon}{2}$ outside the
compact set $B^{+}_{P}(R_{1})$. We conclude that $P^{N}f\in\mathcal{C}_{0}$.
This completes the proof. ∎

###  4.2 Convergence of the generators of geodesic random walks.

In this section, we prove the generators $A_{N}$ converge on the space
$\mathcal{C}^{\infty}_{K}$ to some second order elliptic operator with smooth
coefficients.

Denote

$\displaystyle f_{Y-\mu_{p}}(t)=f\circ\gamma_{Y-\mu_{p}}(t)$ (4.10)

######  Proposition 4.3.

Let $A$ be the differential operator defined by

$\displaystyle
Af(p)\coloneqq\mathrm{d}f(\mu_{p})+\dfrac{1}{2}\int_{T_{p}M}\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}f_{Y-\mu_{p}}(t)\,\nu_{p}(\mathrm{d}Y),\
f\in\mathcal{C}^{2}.$ (4.11)

Then $A$ is a second order positive definite elliptic operator of smooth
coefficients and for each $f\in\mathcal{C}^{\infty}_{K}$

$\displaystyle\lim_{N\to\infty}\lVert A_{N}f-Af\|=0.$ (4.12)

###### 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 $N\rightarrow\infty$.

Take any $f\in\mathcal{C}^{\infty}_{K}$. We have

$\displaystyle A_{N}(f)(p)$ $\displaystyle=N\Big{(}P^{N}(f)(p)-f(p)\Big{)}$
(4.13) $\displaystyle=N\
\int_{T_{p}M}\Big{[}f\circ\gamma_{Y-(1-1/\sqrt{N})\mu_{p}}\Big{(}\dfrac{1}{\sqrt{N}}\Big{)}-f(p)\Big{]}\,\nu_{p}(\mathrm{d}Y)$

Then for any $p\in M$ and $Y\in D_{p}M$, the Taylor expansion of

$\displaystyle
f_{Y-(1-1/\sqrt{N})\mu_{p}}(t)=f\circ\gamma_{Y-(1-1/\sqrt{N})\mu_{p}}(t)$
(4.14)

gives

$\displaystyle f_{Y-(1-1/\sqrt{N})\mu_{p}}\left(\dfrac{1}{\sqrt{N}}\right)=$
$\displaystyle
f(p)+\dfrac{1}{\sqrt{N}}\mathrm{d}f_{p}(Y-(1-1/\sqrt{N})\mu_{p})$
$\displaystyle+\dfrac{1}{2N}\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}\left(f_{Y-(1-1/\sqrt{N})\mu_{p}}(t)\right)+R_{N}(p,Y).$

Thus we have

$\displaystyle A_{N}f(p)$
$\displaystyle=N\int_{T_{p}M}\biggr{\\{}\dfrac{1}{\sqrt{N}}\mathrm{d}f_{p}(Y-(1-1/\sqrt{N})\mu_{p})$
(4.15)
$\displaystyle+\dfrac{1}{2N}\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}(f_{Y-(1-1/\sqrt{N})\mu_{p}}(t))+R_{N}(p,Y)\biggr{\\}}\,\nu_{p}(\mathrm{d}Y)$
$\displaystyle=\mathrm{d}f(\mu_{p})+\int_{T_{p}M}\dfrac{1}{2}\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}(f_{Y-(1-1/\sqrt{N})\mu_{p}}(t))\,\nu_{p}(\mathrm{d}Y)+\int_{T_{p}M}NR_{N}(p,Y)\,\nu_{p}(\mathrm{d}Y).$

Using Lemma 3.1 and equation (4.4), for any $Y\in D_{p}M$ and $p\in M$, there
is some constant $c_{f}>0$ such that

$\displaystyle|R_{N}(p,Y)|$ $\displaystyle\leq\dfrac{1}{N\sqrt{N}}\
\sup_{t\in[0,1/\sqrt{N}]}\
\biggr{\lvert}\dfrac{\mathrm{d}^{3}}{\mathrm{d}t^{3}}f_{Y-(1-1/\sqrt{N})\mu_{p}}(t)\biggr{\rvert}$
(4.16)
$\displaystyle\leq\dfrac{c_{f}}{N\sqrt{N}}F^{3}(Y-(1-1/\sqrt{N})\mu_{p})$
$\displaystyle\leq\dfrac{c_{f}}{N\sqrt{N}}(F(Y)+F(-\mu_{p}))^{3}$
$\displaystyle\leq\dfrac{c_{f}}{N\sqrt{N}}(C+1)^{3}.$

Clearly, we have from (4.16):

$\displaystyle\lim_{N\to\infty}\sup_{p\in M,Y\in D_{p}M}|NR_{N}(p,Y)|=0.$
(4.17)

The last term in (4.15) tends to zero, since $\nu_{p}$ is only supported on
$D_{p}M$.

For the second order term, in a canonical coordinate of $TM$, we have

$\displaystyle\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=0}(f_{Y-(1-1/\sqrt{N})\mu_{p}}(t))$
(4.18) $\displaystyle=f_{ij}\cdot
y^{i}\left(Y-(1-\dfrac{1}{\sqrt{N}})\mu_{p}\right)y^{j}\left(Y-(1-\dfrac{1}{\sqrt{N}})\mu_{p}\right)$
$\displaystyle\quad-
f_{k}\cdot\Gamma^{k}_{ij}\left(p,Y-(1-\dfrac{1}{\sqrt{N}})\mu_{p}\right)y^{i}\left(Y-(1-\dfrac{1}{\sqrt{N}})\mu_{p}\right)y^{j}\left(Y-(1-\dfrac{1}{\sqrt{N}})\mu_{p}\right).$

Since the formal Christoffel symbols are bounded on each compact local
coordinate, the right-hand-side of (LABEL:eqn:second_order_term) converges to

$\displaystyle f_{ij}\cdot
y^{i}(Y-\mu_{p})y^{j}(Y-\mu_{p})-f_{k}\cdot\Gamma^{k}_{ij}(p,Y-\mu_{p})y^{i}(Y-\mu_{p})y^{j}(Y-\mu_{p}),$

as $N\rightarrow\infty$ uniformly on $DK$ for each compact chart $K\subseteq
M$, where $DK=\\{Y\in D_{p}M\colon p\in K\\}$.

Choose a smooth coordinate on some open $U\subseteq M$. The chain rule implies
$A$ has the following form in this coordinate.

$\displaystyle Af(p)=$
$\displaystyle\mathrm{d}f(\mu_{p})+\dfrac{1}{2}\biggr{(}f_{ij}\int_{T_{p}M}y^{i}(Y-\mu_{p})y^{j}(Y-\mu_{p})\,\nu_{p}(\mathrm{d}Y)$
$\displaystyle-
f_{k}\int_{T_{p}M}\Gamma^{k}_{ij}(p,Y-\mu_{p})y^{i}(Y-\mu_{p})y^{j}(Y-\mu_{p})\,\nu_{p}(\mathrm{d}Y)\biggr{)}.$

Because $f$ has compact support, we have

$\displaystyle\lim_{n\rightarrow\infty}\lVert A_{N}f-Af\rVert=0,\quad
f\in\mathcal{C}^{\infty}_{K}.$ (4.19)

It follows that the symbol of $A$ is:

$\displaystyle\sigma(A)(p)=\dfrac{1}{2}\int_{T_{p}M}\otimes^{2}(Y-\mu_{p})\,\nu_{p}(\mathrm{d}Y).$
(4.20)

For each $p\in M$, the measure $\nu_{p}$ is induced by either a smooth non-
zero $(m-1)$-form on $I_{p}M$ or an $m$-form $T_{p}M$ (condition
$\mathbf{H}_{\nu}$). Then $\sigma(A)$ is strictly positive definite, and hence
$A$ is a strictly elliptic operator.

In each compact local coordinate, the functions $\\{\Gamma^{k}_{ij}\\}$ are
bounded, and smooth on $TM\setminus\\{0\\}$. It follows that $A$ has smooth
coefficients. This completes the proof.

∎

###  4.3 Tightness of the family $\\{\xi^{N}\\}$.

In this section we prove the family of random walks $\\{\xi^{N}\\}$ is tight
in $D([0,\infty),M)$. Recall that the symmetrized distance $d$ makes $(M,d)$ a
complete separable metric space, as $(M,F)$ is forward complete and has
bounded geometry.

######  Proposition 4.4.

Let the Finsler manifold $(M,F)$ and the family $\\{\nu_{p}\\}_{p\in M}$
satisfy Assumptions _H_ c, _H_ b and $\mathbf{H}_{\nu}$. Then the family of
random walks $\\{\xi^{N}\\}_{N\geq 1}$ is tight in $D([0,\infty),M)$.

###### 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 $\xi^{N}$, see Equation (4.53). Because $M$ is in general non-
compact, the injective radius lower bound $\operatorname{inj}_{M}$ can be
zero. Since the non-symmetrized distance function $d_{a}(p,\cdot)$ is smooth
only within the injective radius, we work on the tangent bundle $TM$ to bypass
this technical problem. To prepare the proof of Lemma 4.8 as well as Lemma
4.9, for $R\geq 0$, define

$\displaystyle D_{p}(R)\coloneqq\\{Y\in T_{p}M\colon F(Y)\leq R\\}.$

We make the following construction.

The condition $K\leq\lambda$ implies there exists some $0<\delta_{c}<1$ so
that the conjugate radius $\mathrm{con}_{M}>\delta_{c}$, see Proposition 3.2.
For each $p\in M$, the exponential map $\exp_{p}$ is a smooth immersion on
$D_{p}(\delta_{c})$ 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 $D_{p}(\frac{\delta_{c}}{2})$, while $F_{p}$ is
the standard Minkowski metric on $T_{p}M\setminus D_{p}(1)$, under any
standard identification $T_{p}M\simeq\mathbb{R}^{m}$. To distinguish it from
the distance functions on $(M,F)$, we denote the asymmetric and symmetric
distance on $(T_{p}M,F_{p})$ by $d_{a}^{p}$ and $d^{p}$, respectively. Note
that the injective radius of $F_{p}$ at $0\in T_{p}M$ is at least
$\frac{\delta_{c}}{4}$.

Now for each $p\in M$ we construct the measures $\\{\tilde{\nu}_{q}\\}_{q\in
T_{p}M}$, so that on $D_{p}(\delta_{c}/2)$, the measures
$\\{\tilde{\nu_{q}}\\}_{q\in D_{p}(\delta_{c}/2)}$ are the lift of
$\\{\nu_{o}\\}_{o\in M}$ by the exponential map $\exp_{p}$. In addition, we
require the measures $\\{\tilde{\nu}_{q}\\}_{q\in T_{p}M}$ satisfy the
condition $\mathbf{H}_{\nu}$.

Then for each $p\in M$ and $N\geq 1$, we construct an $N$-scaled geodesic
random walk $\xi^{N,p}$ on the Finsler manifold $(T_{p}M,F_{p})$ starting at
$0\in T_{p}M$, using the prescribed measures $\\{\tilde{\nu}_{q}\\}_{q\in
T_{p}M}$ as in Section 3.2. Note as $(T_{p}M,F_{p})$ satisfies
$\mathbf{H}_{b}$ and $\mathbf{H}_{c}$, all results we proved earlier are true
for the random walks $\xi^{N,p}$.

######  Lemma 4.5.

There exists some $\delta_{0}>0$ so that for each $\delta\in(0,\delta_{0})$,
there exists a family of functions $\\{f_{p}^{\delta}\\}_{p\in M}$ such that

  1. 1.

Each $f_{p}^{\delta}$ is a function on $T_{p}M$ with $0\leq f_{p}^{\delta}\leq
1$ such that $f_{p}^{\delta}(0)=1$ and $f^{\delta}_{p}(q)=0$ if $q\notin
D_{p}(\delta)$.

  2. 2.

Denote $A_{N,p}$ the generator associated to $\xi^{N,p}$, then there exists a
constant $\tilde{C}(\delta)>0$ such that

$\displaystyle\sup_{N\geq 1}\sup_{p\in M}\sup_{q\in
T_{p}M}\Big{|}A_{N,p}f_{p}^{\delta}(q)\Big{|}\leq\tilde{C}(\delta).$ (4.21)

###### Proof.

The general scheme to prove this lemma is as follows. We construct the family
of functions $\\{f^{\delta}_{p}\\}$ using the distance functions
$d_{a}^{p}(0,\cdot)$ 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 the distance functions have uniformly bounded Hessians.
This fact applied to the Taylor expansion of $f^{\delta}_{p}$ implies the
family of functions we constructed satisfies the conditions listed in the
Lemma.

For $\delta_{c}\in(0,1)$ chosen above, $C>0$ and $\lambda>0$ from Definition
2.1, let
$0<\delta_{0}<\min\\{\frac{\delta_{c}}{4(C+1)},\frac{\pi}{2\sqrt{\lambda}}\\}$.
Fix any $p\in M$, and denote $d_{a}^{p}(0,\cdot)$ the distance function for
$(T_{p}M,F_{p})$ from $0\in T_{p}M$. Because the injective radius for $F_{p}$
at $0$ is at least $\frac{\delta_{c}}{4}$, the distance function
$d_{a}^{p}(0,\cdot)$ is smooth on the open set $D_{p}(\delta)\setminus\\{0\\}$
for each $0<\delta<\delta_{0}$.

Fix any $\delta\in(0,\delta_{0})$. Let
$\psi^{\delta}\colon\mathbb{R}\to\mathbb{R}$ be a smooth function with compact
support contained in $[-\frac{\delta}{2},\frac{\delta}{2}]$. Further suppose
$0\leq\psi^{\delta}\leq 1$ and $\psi^{\delta}\equiv 1$ on
$I_{1}=[-\frac{\delta}{4},\frac{\delta}{4}]$. For each $p\in M$ , the function

$\displaystyle f_{p}^{\delta}(q):=\psi^{\delta}\circ d_{a}^{p}(0,q),\quad q\in
T_{p}M.$ (4.22)

is smooth on $T_{p}M$ and satisfies condition 1.

To prove condition 2, for any $q\in(T_{p}M,F_{p})$, let $\tilde{\mu}_{q}$ be
the mean of $\tilde{\nu}_{q}$. An argument similar to Proposition 4.2 shows
that for any $q\in T_{p}M$

$\displaystyle
A_{N,p}f_{p}^{\delta}(q)=N\Big{[}\int_{T_{q}(T_{p}M)}f_{p}^{\delta}\circ\gamma_{Y-(1-1/\sqrt{N})\tilde{\mu}_{q}}\Big{(}\frac{1}{\sqrt{N}}\Big{)}\,\tilde{\nu}_{q}(\mathrm{d}Y)-f_{p}^{\delta}(q)\Big{]}.$
(4.23)

Note $\\{\tilde{\nu}_{q}\\}_{q\in T_{p}M}$ satisfies $\mathbf{H}_{\nu}$, so we
only need to integrate over $Y\in T_{q}(T_{p}M)$ with $F_{p}(Y)\leq 1$.

To simplify the notations, define for $Y\in T_{q}(T_{p}M)$

$\displaystyle h_{N}(Y)(t)$
$\displaystyle:=d_{a}\Big{(}0,\gamma_{Y-(1-1/\sqrt{N})\mu_{q}}(t)\Big{)},$
(4.24) $\displaystyle h_{N}^{\delta}(Y)(t)$ $\displaystyle:=\psi^{\delta}\circ
h_{N}(Y)(t)=f_{p}^{\delta}\circ\gamma_{Y-(1-1/\sqrt{N})\mu_{q}}(t),\quad t\geq
0.$ (4.25)

By Taylor theorem, there exist functions $\\{t_{N}\\}$ with

$\displaystyle t_{N}\colon
T_{q}(T_{p}M)\rightarrow\Big{(}0,\frac{1}{\sqrt{N}}\Big{)}$ (4.26)

such that

$\displaystyle
h_{N}^{\delta}(Y)\Big{(}\frac{1}{\sqrt{N}}\Big{)}=f_{p}^{\delta}(q)+\mathrm{d}f_{p}^{\delta}(q)\Big{(}\frac{Y-\tilde{\mu}_{q}}{\sqrt{N}}+\frac{1}{N}\tilde{\mu}_{q}\Big{)}+\frac{1}{2N}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=t_{N}(Y)}\left(h_{N}^{\delta}(Y)(t)\right).$
(4.27)

Using Equations (4.23) and (4.27), we get

$\displaystyle A_{N}f_{p}^{\delta}(q)$
$\displaystyle=N\int_{T_{q}(T_{p}M)}\Big{[}\mathrm{d}f_{p}^{\delta}(q)\left(\dfrac{Y-\tilde{\mu}_{q}}{\sqrt{N}}+\dfrac{1}{N}\tilde{\mu}_{q}\right)+\dfrac{1}{2N}\dfrac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=t_{N}(Y)}\left(h_{N}^{\delta}(Y)(t)\right)\,\Big{]}\tilde{\nu}_{q}(\mathrm{d}Y)$
$\displaystyle=\mathrm{d}f^{\delta}_{p}(q)(\tilde{\mu}_{q})+\frac{1}{2}\int_{T_{q}(T_{p}M)}\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=t_{N}(Y)}\left(h_{N}^{\delta}(Y)(t)\right)\tilde{\nu}_{q}(\mathrm{d}Y).$
(4.28)

We need to show the equation above is uniformly bounded for all $p\in M$,
$q\in T_{p}M$ and $N\geq 1$.

First we show for each $0<\delta<\delta_{0}$,

$\displaystyle\sup_{p\in M}\sup_{q\in
T_{p}M}\lvert\mathrm{d}f^{\delta}_{p}(q)(\tilde{\mu}_{q})\rvert<\infty.$
(4.29)

Clearly if $d_{a}^{p}(0,q)\leq\frac{\delta}{4}$ or
$d_{a}^{p}(0,q)\geq\frac{\delta}{2}$, we have

$\displaystyle\mathrm{d}f^{\delta}_{p}(q)(\tilde{\mu}_{q})=0.$

For any $q\in T_{p}M$ such that $\frac{\delta}{4}\leq
d_{a}(0,q)\leq\frac{\delta}{2}$, the Finlser metric $F_{p}|_{T_{q}(T_{p}M)}$
and the measure $\tilde{\nu}_{q}$ are the pull backs of $F$ and
$\\{\nu_{o}\\}_{o\in M}$ by $\exp_{p}$, respectively. Thus
$F_{p}(\tilde{\mu}_{q})<1$ and $F_{p}(-\tilde{\mu}_{q})\leq C$. It follows
that

$\displaystyle|\tilde{\mu}_{q}(d^{p}_{a}(0,\cdot))|\leq C\quad\mathrm{if}\
\frac{\delta}{4}\leq d_{a}^{p}(0,q)\leq\frac{\delta}{2}.$

The function $d_{a}^{p}(0,\cdot)$ is smooth at $q$ For $q\in T_{p}M$ with
$\frac{\delta}{4}\leq d_{a}(0,q)\leq\frac{\delta}{2}$. Hence Equation (4.29)
holds by the chain rule.

Now it suffices to prove the integrand in (4.28) is uniformly bounded for all
$Y\in T_{q}(T_{p}M)$, $q\in T_{p}M$, $p\in M$ and $N\geq 1$. By the
construction of $\psi^{\delta}$, for the case

$\displaystyle
d_{a}^{p}\Big{(}0,\gamma_{Y-(1-1/\sqrt{N})\tilde{\mu}_{q}}(t_{N}(Y))\Big{)}\geq\frac{\delta}{2}\quad\mathrm{or}\quad
d_{a}^{p}\Big{(}0,\gamma_{Y-(1-1/\sqrt{N})\tilde{\mu}_{q}}(t_{N}(Y))\Big{)}\leq\frac{\delta}{4},$
(4.30)

we have

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=t_{N}(Y)}h_{N}^{\delta}(Y)(t)=0.$
(4.31)

For the case

$\displaystyle\frac{\delta}{4}\leq
d_{a}^{p}\big{(}0,\gamma_{Y-(1-1/\sqrt{N})\tilde{\mu}_{q}}(t_{N}(Y))\big{)}\leq\frac{\delta}{2},$
(4.32)

the function $h_{N}(Y)(t)$ is smooth on some interval containing $t=t_{N}(Y)$,
because $d_{a}^{p}(0,\cdot)$ is smooth on $D_{p}(\delta)\setminus\\{0\\}$.
Since $\psi^{\delta}$ is in $\mathcal{C}^{\infty}_{K}$, 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 $\nabla\rho:=\nabla d_{a}^{p}(0,\cdot)$
the _Finsler gradient_ , see, e.g. [38, Equation (3.14) in §3.2]. Following
[38, §15.1], let us define

$\displaystyle\hat{\mathbf{g}}\coloneqq\tilde{g}_{\nabla\rho},\quad
r_{N}(Y)\coloneqq
d_{a}^{p}\Big{(}0,\gamma_{Y-(1-1/\sqrt{N})\tilde{\mu}_{q}}(t_{N})\Big{)},$
$\displaystyle\dot{\gamma}_{N}\coloneqq\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=t_{N}(Y)}\gamma_{Y-(1-1/\sqrt{N})\tilde{\mu}_{q}}(t),\quad
Y^{\perp}\coloneqq\dot{\gamma}_{N}-\hat{\mathbf{g}}(\dot{\gamma}_{N},\nabla\rho)\nabla\rho.$

Here $\tilde{g}$ is the fundamental tensor of $F_{p}$.

Because $\exp_{p}$ is an isometric immersion on $D_{p}(\delta)$, on
$(T_{p}M,F_{p})$, we also have the uniform elliptic conditions

$\displaystyle\dfrac{1}{C^{2}}\tilde{g}_{v}(v,v)\leq\tilde{g}_{u}(v,v)\leq
C^{2}\tilde{g}_{v}(v,v),$ (4.33)

for any $0\neq u,v\in T_{q}(T_{p}M)$ with $q\in D_{p}(\delta)$. It follows
that for all $Y\in T_{q}(T_{p}M)$ with $F_{p}(Y)\leq 1$, $q\in D_{p}(\delta)$
we have

$\displaystyle
F_{p}(\dot{\gamma}_{N})=F_{p}\Big{(}Y-(1-1/\sqrt{N})\tilde{\mu}_{q}\Big{)}\leq
C+1,$ $\displaystyle F_{p}(-\dot{\gamma}_{N})\leq C+1.$

This implies that for all $Y\in T_{q}(T_{p}M)$ with $F_{p}(Y)\leq 1$,

$\displaystyle\Big{|}\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=t_{N}(Y)}h_{N}(Y)(t)\Big{|}\leq
C+1,$

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 $(D_{p}(\delta),F_{p})$ also has flag
curvature and $T$-curvature bounded by $|K|\leq\lambda$ and $|T|\leq\lambda$,
because $\exp_{p}$ restricted to $(D_{p}(\delta),F_{p})$ is an isometric
immersion. Since the injective radius of $F_{p}$ at $0$ is at least
$\frac{\delta_{c}}{4}>\delta$, the Hessian comparison theorem implies:

$\displaystyle\left(\sqrt{\lambda}\cdot\cot(\sqrt{\lambda}\cdot
r(Y))-\lambda\right)\hat{\mathbf{g}}(Y^{\perp},Y^{\perp})\leq\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\biggr{|}_{t=t_{N}(Y)}h_{N}(Y(t)),$
(4.34)
$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\Big{|}_{t=t_{N}(Y)}h_{N}(Y(t))\leq\left(\sqrt{\lambda}\cdot\coth(\sqrt{\lambda}\cdot
r(Y))+\lambda\right)\hat{\mathbf{g}}(Y^{\perp},Y^{\perp}).$ (4.35)

Using the fact
$\hat{\mathbf{g}}(\nabla\rho,\nabla\rho)=F^{2}_{p}(\nabla\rho)=1$ on
$D_{p}(\delta)\setminus\\{0\\}$, we get

$\displaystyle\hat{\mathbf{g}}(Y^{\perp},Y^{\perp})$ $\displaystyle\
=\left\lvert\hat{\mathbf{g}}(\dot{\gamma}_{N},\dot{\gamma}_{N})-\hat{\mathbf{g}}^{2}(\dot{\gamma}_{N},\nabla\rho)\right\rvert.$
(4.36)

If $x\in D_{p}(\delta)$, for any tangent vectors $Y_{1},Y_{2}\in
T_{x}(T_{p}M)$ with $Y_{1}\neq 0$, the fundamental inequality in Finsler
geometry (see 1.2.16 of [5]) and the inequality $F_{p}(Y_{2})\leq
CF_{p}(-Y_{2})$ give

$\displaystyle|\tilde{g}_{Y_{1}}(Y_{1},Y_{2})|\leq CF_{p}(Y_{1})F_{p}(Y_{2}).$
(4.37)

Substituting this into (4.36) and using uniform ellipticity, for $Y$ such that
$F_{p}(Y)\leq 1$ and (4.32) holds, we obtain

$\displaystyle\hat{\mathbf{g}}(Y^{\perp},Y^{\perp})$ $\displaystyle\leq
C\tilde{g}_{\dot{\gamma}_{N}}(\dot{\gamma}_{N},\dot{\gamma}_{N})+C^{2}F^{2}_{p}(\dot{\gamma}_{N})$
$\displaystyle\leq
2C^{2}F_{p}^{2}\Big{(}Y-\Big{(}1-\frac{1}{\sqrt{N}}\Big{)}\tilde{\mu}_{q}\Big{)}$
$\displaystyle\leq 2C^{2}(C+1)^{2}.$

Since $\frac{\delta}{4}\leq r_{N}(Y)\leq\frac{\delta}{2}$ and
$\delta<\frac{\pi}{2\sqrt{\lambda}}$, we have for all $N\geq 1$:

$\displaystyle 0\leq\cot(\sqrt{\lambda}\cdot
r_{N}(Y))\leq\coth(\sqrt{\lambda}\cdot r_{N}(Y)).$ (4.38)

Then for all $N\geq 1$ and $Y$ with $F_{p}(Y)\leq 1$, we have the estimate:

$\displaystyle\left\lvert\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}\Big{|}_{t=t_{N}(Y)}h_{N}(Y(t))\right\rvert$
$\displaystyle\leq\Big{(}\sqrt{\lambda}\cdot\coth(\frac{\delta\sqrt{\lambda}}{4})+\lambda\Big{)}\hat{\mathbf{g}}(Y^{\perp},Y^{\perp}),$
(4.39) $\displaystyle\leq
2C^{2}(C+1)^{2}\Big{(}\sqrt{\lambda}\cdot\coth(\frac{\delta\sqrt{\lambda}}{4})+\lambda\Big{)}.$
(4.40)

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
some $\tilde{C}(\delta)>0$ such that condition 2 holds. This completes the
proof. ∎

The family of functions $\\{f^{\delta}_{p}\\}$ will be used now to estimate
the first exit time from a $\delta$-balls of the geodesic random walk
$\xi^{N}$.

For each $p\in M$, $N\geq 1$ and $\delta>0$, define the following stopping
times for the random walk $\xi^{N}$ on $M$ and $\xi^{N,p}$ on $T_{p}M$:

$\displaystyle\tau^{N,\delta}$ $\displaystyle\coloneqq\inf\\{t>0\colon
d(\xi^{N}_{t},p)>\delta\\},$ (4.41) $\displaystyle\tau^{N,\delta}_{p}$
$\displaystyle\coloneqq\inf\\{t>0\colon d^{p}(\xi^{N,p}_{t},0)>\delta\\},\quad
0\in T_{p}M.$ (4.42)

Now we compare the exit time probabilities of the $\delta$-balls for
$\xi^{N}_{t}$ and $\xi^{N,p}_{t}$ for sufficiently large $N$.

######  Lemma 4.6.

For any $p\in M$, $N\geq 1$ and $\delta$ such that $0<\delta<\delta_{0}$ and
$\frac{2(C+1)}{\sqrt{N}}<\frac{\delta_{c}}{4}$, we have

$\displaystyle\mathbf{P}_{p}(\tau^{N,\delta}\leq
t)\leq\mathbf{P}_{0}(\tau^{N,\delta}_{p}\leq t),\quad\forall t\geq 0.$ (4.43)

###### Proof.

The geodesic random walks $\xi^{N,p}$ and $\xi^{N}$ are constructed by
randomizing the time of the discrete Markov processes $\zeta^{N,p}$ and
$\zeta^{N}$ using a Poisson process, respectively. Hence it suffices to show
for each pair $(N,\delta)$ satisfies the condition in the lemma, the following
holds

$\displaystyle\mathbf{P}_{0}\left(\max_{j\leq
k}d^{p}\left(0,\zeta_{j}^{N,p}\right)\leq\delta\right)\leq\mathbf{P}_{p}\left(\max_{j\leq
k}d(p,\zeta^{N}_{j})\leq\delta\right),\ \forall p\in M,\
\forall\delta<\delta_{0},\ k\geq 0.$ (4.44)

For $r>0$, define the closed $\delta$-balls of the symmetrized distances on
$T_{p}M$ and $M$ respectively:

$\displaystyle B_{0}^{p}(r)$ $\displaystyle=\\{y\in T_{p}M\colon
d^{p}(0,y)\leq r\\},$ $\displaystyle B_{p}(r)$ $\displaystyle=\\{q\in M\colon
d(p,q)\leq r\\}.$

Because $\delta<\delta_{0}<\frac{\delta_{c}}{4(C+1)}$, we have
$B^{p}_{0}(\delta)\subset D_{p}(\delta_{c}/2)$. Hence $\exp_{p}$ maps
$(B^{p}_{0}(\delta),F_{p})$ inside $(B_{p}(\delta),F)$ by an isometric
immersion. Now for each $k\geq 0$ define the following Borel sub-probabilty
measures on $B^{p}_{0}(\delta)$ and $B_{p}(\delta)$, respectively.

$\displaystyle\theta^{0}_{k}(\hat{E})$
$\displaystyle=\mathbf{P}_{0}\left(\zeta^{N,p}_{j}\in B^{p}_{0}(\delta),\
1\leq j\leq k-1,\ \zeta^{N,p}_{k}\in\hat{E}\right),\
\forall\hat{E}\in\mathcal{B}(B^{p}_{0}(\delta));$ $\displaystyle\theta_{k}(E)$
$\displaystyle=\mathbf{P}_{p}\left(\zeta^{N}_{j}\in B_{p}(\delta),\ 1\leq
j\leq k-1,\ \zeta^{N}_{k}\in E\right),\ \forall
E\in\mathcal{B}(B_{p}(\delta)).$

Let
$\theta^{p}_{k}\coloneqq\left(\exp_{p}|_{B^{p}_{0}(\delta)}\right)_{*}\theta^{0}_{k}$.
For integers $N$ such that $\frac{2(C+1)}{\sqrt{N}}<\frac{\delta_{c}}{4}$, we
claim $\theta^{p}_{k}\leq\theta_{k}$ for all $k\geq 0$.

We prove the claim by induction. As for $k=0$, we have
$\theta^{0}_{0}(\hat{E})=\mathbf{1}_{\hat{E}}(0)$ and
$\theta_{0}(E)=\mathbf{1}_{E}(p)$. Since $\exp_{p}(0)=p$, this claim holds for
$k=0$. For simplicity we denote

$\displaystyle\mathbf{e}_{p}^{N}\coloneqq\exp_{p}\Big{(}\frac{Y-\mu_{p}}{\sqrt{N}}+\frac{1}{N}\mu_{p}\Big{)}.$
(4.45)

For any $q\in B^{p}_{0}(\delta)$ and $Y\in T_{q}(T_{p}M)$ with $F_{p}(Y)\leq
1$, the condition $\frac{2(C+1)}{\sqrt{N}}<\frac{\delta_{c}}{4}$ implies the
$N$-scaled geodesic segment $\gamma(t)=\exp_{q}(t\mathbf{e}^{N}_{q}(Y))$ for
$t\in[0,1]$ of $F_{p}$ is mapped by $\exp_{p}$ to a geodesic segment on
$(M,F)$. Also note for $q\in B^{p}_{0}(\delta)$, the mean is preserved under
$\exp_{p}$ by

$\displaystyle(\mathrm{d}\exp_{p}(q))_{*}(\tilde{\mu}_{q})=\mu_{\exp_{p}(q)}.$

Hence for any $q\in B^{p}_{0}(\delta)$ and $Y\in T_{q}(T_{p}M)$ with
$F_{p}(Y)\leq 1$ we get

$\displaystyle\exp_{p}\left(\mathbf{e}^{N}_{q}(Y)\right)=\mathbf{e}^{N}_{\exp_{p}(q)}\left((\mathrm{d}\exp_{p}(q))_{*}(Y)\right).$
(4.46)

For each $N\geq 1$, let $P^{o}(x,\cdot)$ and $P(y,\cdot)$ be the one-step
transition probabilities of $\zeta^{N,p}$ and $\zeta^{N}$, respectively. For
$q\in B^{p}_{0}(\delta)$, set $q_{1}=\exp_{p}(q)$. Since $\tilde{\nu}_{q}$ is
the pull-back of $\nu_{q_{1}}$ by $(d\exp_{p}(q))$, then (4.46) implies for
any Borel set $\hat{E}\subset B^{p}_{0}(\delta)$, we have

$\displaystyle
P^{o}(q,\hat{E})=\tilde{\nu}_{q}\left(\left(\mathbf{e}^{N}_{q}\right)^{-1}(\hat{E})\right)\leq\nu_{q_{1}}\left(\left(\mathbf{e}^{N}_{q_{1}}\right)^{-1}(\exp_{p}(\hat{E}))\right)=P(q_{1},\exp_{p}(\hat{E})).$
(4.47)

The inequality in the previous formula appears because the exponential map
$\exp_{p}$ restricted to $B^{p}_{0}(\delta)$ is not necessarily injective.

In particular, for any Borel $E\subset B_{p}(\delta)$, we get

$\displaystyle P(q_{1},E)\geq P^{o}(q,(\exp_{p})^{-1}(E)\cap
B^{p}_{0}(\delta)).$ (4.48)

By the Markov property, we have for all $k\geq 1$

$\displaystyle\theta^{o}_{k}(\hat{E})$
$\displaystyle=\int_{B^{p}_{0}(\delta)}P^{o}(y,\hat{E})\theta^{o}_{k-1}(\mathrm{d}y),\
\forall\hat{E}\in\mathcal{B}(B^{p}_{0}(\delta));$ $\displaystyle\theta_{k}(E)$
$\displaystyle=\int_{B_{p}(\delta)}P(x,E)\theta_{k-1}(\mathrm{d}x),\ \forall
E\in\mathcal{B}(B_{p}(\delta)).$

Hence we have the following chain of inequalities:

$\displaystyle\theta_{k}(E)$
$\displaystyle=\int_{B_{p}(\delta)}P(x,E)\theta_{k-1}(\mathrm{d}x)$
$\displaystyle\geq\int_{B_{p}(\delta)}P(x,E)\theta^{p}_{k-1}(\mathrm{d}x)$
$\displaystyle=\int_{B^{p}_{0}(\delta)}P(\exp_{p}(y),E)\theta^{o}_{k-1}(\mathrm{d}y)$
$\displaystyle\geq\int_{B^{p}_{0}(\delta)}P^{o}(y,(\exp_{p})^{-1}(E)\cap
B^{p}_{0}(\delta))\theta^{o}_{k-1}(\mathrm{d}y)$
$\displaystyle=\theta^{p}_{k}(E).$

where the first inequality comes for the induction assumption, and the second
inequality is due to (4.48).

In particular, for all $k\geq 0$, the inequality

$\displaystyle\theta^{o}_{k}(B^{p}_{0}(\delta))=\theta^{p}_{k}(B_{p}(\delta))\leq\theta_{k}(B_{p}(\delta))$

implies (4.44) holds. This completes the proof. ∎

Next, we prove the following estimate on the first exit times of $\xi^{N}$
from $\delta$-balls on $M$.

######  Lemma 4.7.

For each $\delta>0$ there is $C(\delta)>0$ such that for all $t\geq 0$

$\displaystyle\sup_{p\in M}\sup_{N\geq 1}\mathbf{P}_{p}(\tau^{N,\delta}\leq
t)\leq C(\delta)t.$ (4.49)

In particular,

$\displaystyle\sup_{p\in M}\sup_{N\geq
1}\mathbf{E}_{p}\mathrm{e}^{-\tau^{N,\delta}}<1.$ (4.50)

###### 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 $\xi^{N,p}$, and we
show

$\displaystyle\sup_{p\in M}\sup_{N\geq
1}\mathbf{P}_{0}(\tau^{N,\delta}_{p}\leq t)\leq\tilde{C}(\delta)t,$ (4.51)

where $0\in T_{p}M$ and $\tilde{C}(\delta)>0$ 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).

It suffices to show (4.51) for sufficiently small $\delta$. For any
$\delta\in(0,\delta_{0})$ and $p\in M$, let $\\{f^{\delta}_{p}\\}$ be the
family of functions constructed in Lemma 4.5. Using $0\leq f^{\delta}_{p}\leq
1$ and $f^{\delta}_{p}(0)=1$, we have

$\displaystyle\mathbf{P}_{0}(\tau^{N,\delta}_{p}\leq t)$
$\displaystyle=1-\mathbf{P}_{0}(\tau^{N,\delta}_{p}>t)$ $\displaystyle\leq
1-\mathbf{E}_{0}\Big{[}\mathbb{I}(\tau^{N,\delta}_{p}>t)f^{\delta}_{p}(\xi^{N,p}(\tau^{N,\delta}_{p}\wedge
t))\Big{]},$
$\displaystyle=1-\mathbf{E}_{0}\Big{[}\Big{(}1-\mathbb{I}(\tau^{N,\delta}_{p}\leq
t)\Big{)}f^{\delta}_{p}\Big{(}\xi^{N,p}(\tau^{N,\delta}_{p}\wedge
t)\Big{)}\Big{]}$
$\displaystyle=f^{\delta}_{p}(0)-\mathbf{E}_{0}f^{\delta}_{p}\Big{(}\xi^{N,p}(\tau^{N,\delta}_{p}\wedge
t)\Big{)}+\mathbf{E}_{0}\Big{[}\mathbb{I}(\tau^{N,\delta}_{p}\leq
t)f^{\delta}_{p}\Big{(}\xi^{N,p}(\tau^{N,\delta}_{p}\wedge t)\Big{)}\Big{]}$
$\displaystyle=f^{\delta}_{p}(0)-\mathbf{E}_{0}f^{\delta}_{p}\Big{(}\xi^{N,p}(\tau^{N,\delta}_{p}\wedge
t)\Big{)}+\mathbf{E}_{0}\Big{[}\mathbb{I}(\tau^{N,\delta}_{p}\leq
t)f^{\delta}_{p}\Big{(}\xi^{N,p}(\tau^{N,\delta}_{p})\Big{)}\Big{]}.$

Here the notation $a\wedge b:=\min\\{a,b\\}$ is standard in the theory of
stochastic processes. Taking into account that
$f^{\delta}_{p}\left(\xi^{N,p}\left(\tau^{N,\delta}_{p}\right)\right)=0$ and
applying the Dynkin formula to above, we obtain

$\displaystyle\mathbf{P}_{0}(\tau^{N,\delta}_{p}\leq
t)\leq-\mathbf{E}_{0}\int^{\tau^{N,\delta}_{p}\wedge
t}_{0}A_{N,p}f^{\delta}_{p}(\xi^{N,p}_{s})\,\mathrm{d}s\leq\sup_{N\geq
1}\lVert A_{N,p}f^{\delta}_{p}\rVert\cdot t=\tilde{C}(\delta)t,\ \forall p\in
M.$

Let $N_{0}$ be the smallest positive integer such that
$\dfrac{2(C+1)}{\sqrt{N_{0}}}<\dfrac{\delta_{c}}{4}$. For $N\leq N_{0}$, we
have always have

$\displaystyle\mathbf{P}_{p}(\tau^{N,\delta}\leq t)\leq\mathbf{P}(Q(Nt)>0)\leq
Nt\leq N_{0}t.$ (4.52)

This together with Lemma 4.6 proves the inequality (4.49) by setting
$C(\delta)=\max\\{\tilde{C}(\delta),N_{0}\\}$.

Furthermore, for $t_{*}=\dfrac{1}{2C(\delta)}>0$

$\displaystyle\mathbf{E}_{p}\mathrm{e}^{-\tau^{N,\delta}}$
$\displaystyle=\mathbf{E}_{p}\mathbb{I}(\tau^{N,\delta}\leq
t_{*})\mathrm{e}^{-\tau^{N,\delta}}+\mathbf{E}_{p}\mathbb{I}(\tau^{N,\delta}>t_{*})\mathrm{e}^{-\tau^{N,\delta}}$
$\displaystyle\leq\mathbf{P}_{p}(\tau^{N,\delta}\leq
t_{*})+\mathrm{e}^{-t_{*}}\left(1-\mathbf{P}_{p}(\tau^{N,\delta}\leq
t_{*})\right)$
$\displaystyle=\mathrm{e}^{-t_{*}}+(1-\mathrm{e}^{-t_{*}})\mathbf{P}_{p}(\tau^{N,\delta}\leq
t_{*}).$

Note that due to (4.49), we have $\mathbf{P}_{p}(\tau^{N,\delta}\leq
t_{*})\leq C(\delta)t_{*}\leq\dfrac{1}{2}$. Hence we obtain

$\displaystyle\mathbf{E}_{p}\mathrm{e}^{-\tau^{N,\delta}}\leq\mathrm{e}^{-t_{*}}+\dfrac{1-\mathrm{e}^{-t_{*}}}{2}=\dfrac{1+\mathrm{e}^{-t_{*}}}{2}<1.$

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 $p\in M$, any $T>0$, $\delta>0$, and any
$(\mathcal{F}^{N})$-stopping times $0\leq\tau\leq T$, we have

$\displaystyle\lim_{s\rightarrow
0}\limsup_{N\rightarrow\infty}\sup_{\tau}\sup_{h\in[0,s]}\mathbf{P}_{p}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta\right)=0$
(4.53)

###### Proof.

Let $\delta,s>0$ be fixed. For each $N\geq 1$, $p\in M$, $h\in[0,s]$ and a
stopping time $\tau$ we have

$\displaystyle\mathbf{P}_{p}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta\right)$
$\displaystyle=\mathbf{E}_{p}\biggr{[}\mathbb{I}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta)\right)\biggr{]},$
$\displaystyle=\mathbf{E}_{p}\biggr{[}\mathbf{E}\biggr{[}\mathbb{I}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta)\right)|\mathcal{F}^{N}_{\tau}\biggr{]}\biggr{]},$
$\displaystyle=\mathbf{E}_{p}\biggr{[}\mathbf{P}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta|\mathcal{F}^{N}_{\tau}\right)\biggr{]}.$

The strong Markov property of $\xi^{N}$ yields

$\displaystyle\mathbf{E}_{p}\biggr{[}\mathbf{P}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta|\mathcal{F}^{N}_{\tau}\right)\biggr{]}=\mathbf{E}_{p}\mathbf{P}_{\xi^{N}_{\tau}}\biggr{(}d(\xi^{N}_{0},\xi^{N}_{h})>\delta\biggr{)}\leq\mathbf{E}_{p}\Big{[}\sup_{q\in
M}\mathbf{P}_{q}(\tau^{N,\delta}\leq h)\Big{]}.$

Thus by Lemma 4.7, we have

$\displaystyle\mathbf{P}_{p}\left(d(\xi^{N}_{\tau},\xi^{N}_{\tau+h})>\delta\right)\leq
C(\delta)h\leq C(\delta)s.$ (4.54)

Taking supremums and letting $s\rightarrow 0$, we obtain the limit in Equation
(4.53). ∎

Next we show the family of processes $\\{\xi^{N}\\}$ has compact containment
property as follows.

######  Lemma 4.9 (compact containment condition).

For any $\varepsilon>0$, $T\geq 0$ and $p\in M$ there is a compact
neighborhood $K_{\varepsilon}(p)\subseteq M$ of $p$ such that

$\displaystyle\inf_{N}\mathbf{P}_{p}\Big{(}\xi^{N}_{t}\in K_{\varepsilon}(p),\
t\in[0,T]\Big{)}\geq 1-\varepsilon.$ (4.55)

###### Proof.

Let us define the following sequence of exit times

$\displaystyle\tau^{N}_{0}\ $ $\displaystyle\coloneqq 0,$
$\displaystyle\tau^{N}_{k}$
$\displaystyle\coloneqq\inf\left\\{s>\tau^{N}_{k-1}\colon
d\left(\xi^{N}_{s},\xi^{N}_{\tau^{N}_{k-1}}\right)>1\right\\},\ k\geq 1,$

(as usual, we set $\inf\emptyset=+\infty$).

By Lemma 4.7, there exists some constant $c_{1}\in(0,1)$ such that

$\displaystyle\sup_{p\in
M}\sup_{N}\mathbf{E}_{p}\mathrm{e}^{-\tau^{N}_{1}}=\sup_{p\in
M}\sup_{N}\mathbf{E}_{p}\mathrm{e}^{-\tau^{N,\delta_{1}}}\leq c_{1}<1$

Then for $k\geq 1$ and $\forall N\geq 1$, the strong Markov property yields

$\displaystyle\mathbf{E}_{p}\mathrm{e}^{-\tau^{N}_{k}}$
$\displaystyle=\mathbf{E}_{p}\big{[}\mathrm{e}^{-\tau^{N}_{k-1}}\cdot\mathrm{e}^{\tau^{N}_{k-1}-\tau^{N}_{k}}\big{]},$
$\displaystyle=\mathbf{E}_{p}\biggr{[}\mathrm{e}^{-\tau^{N}_{k-1}}\cdot\mathbf{E}\big{[}\mathrm{e}^{\tau^{N}_{k-1}-\tau^{N}_{k}}|\mathcal{F}^{N}_{\tau^{N}_{k-1}}\big{]}\biggr{]},$
$\displaystyle=\mathbf{E}_{p}\biggr{[}\mathrm{e}^{-\tau^{N}_{k-1}}\cdot\mathbf{E}_{\xi^{N}_{\tau^{N}_{k-1}}}\mathrm{e}^{-\tau^{N}_{1}}\biggr{]},$
$\displaystyle\leq c_{1}\cdot\mathbf{E}_{p}\mathrm{e}^{-\tau^{N}_{k-1}}\leq
c_{1}^{k}.$

For any $\varepsilon>0$ and $T\geq 0$, define

$\displaystyle
k_{\varepsilon}\coloneqq\left\lceil\dfrac{\ln{\varepsilon}-T}{\ln{c_{1}}}\right\rceil.$
(4.56)

Then the exponential Markov inequality gives $\forall p\in M$, $\forall N\geq
1$:

$\displaystyle\mathbf{P}_{p}\left(\tau^{N}_{k_{\varepsilon}}\leq
T\right)=\mathbf{P}_{p}\left(\mathrm{e}^{-\tau^{N}_{k_{\varepsilon}}}\geq\mathrm{e}^{-T}\right)\leq\mathrm{e}^{T}\mathbf{E}_{p}\mathrm{e}^{-\tau^{N}_{k_{\varepsilon}}}\leq\mathrm{e}^{T}c_{1}^{k_{\varepsilon}}\leq\varepsilon.$
(4.57)

By construction and the triangle inequality we have that for each $k\geq 1$,
each $N\geq 1$

$\displaystyle
d\left(\xi^{N}_{\tau^{N}_{k}},\xi^{N}_{\tau^{N}_{k-1}}\right)\leq
1+\sup_{N}\sup_{p}d(p,\zeta^{N}_{1}).$ (4.58)

We estimate the last term:

$\displaystyle d(p,\zeta^{N}_{1})$ $\displaystyle\leq\sup_{Y\in
D_{p}M}d\Big{(}p,\exp_{p}\Big{(}\frac{Y-\mu_{p}}{\sqrt{N}}+\frac{\mu_{p}}{N}\Big{)}\Big{)}$
(4.59) $\displaystyle\leq\sup_{Y\in D_{p}M}\dfrac{C}{\sqrt{N}}{}\cdot
F\Big{(}Y-\Big{(}1-\frac{1}{\sqrt{N}}\Big{)}\mu_{p}\Big{)}$ (4.60)
$\displaystyle\leq C(C+1).$ (4.61)

Thus we have for $k\geq 1$

$\displaystyle d\left(\xi^{N}_{0},\xi^{N}_{\tau^{N}_{k}}\right)\leq
k\Big{(}1+C(C+1)\Big{)}.$ (4.62)

Now for $p\in M$, $\varepsilon>0$, and $k_{\varepsilon}$ defined as in
Equation (4.56), consider the closed ball

$\displaystyle K_{p}(\varepsilon)\coloneqq\\{q\in M\colon d(p,q)\leq
R(\varepsilon,T)\\},$ (4.63)

with radius

$\displaystyle R(\varepsilon,T)=k_{\varepsilon}(1+C(C+1))+1.$ (4.64)

Then $K_{p}(\varepsilon)$ is closed and forward bounded, hence it is compact
by Hopf–Rinow theorem.

Eventually we get that $\forall p\in M$ and $N\geq 1$:

$\displaystyle\mathbf{P}_{p}$ $\displaystyle\left(\xi^{N}_{t}\notin
K_{p}(\varepsilon)\ \mathrm{for\ some\ }t\leq T\right)$ (4.65)
$\displaystyle\leq\mathbf{P}_{p}\left(\tau^{N}_{k_{\varepsilon}}\leq
T\right)+\mathbf{P}_{p}\left(\xi^{N}_{t}\notin K_{p}(\varepsilon)\
\mathrm{for\ some\ }t\leq T,\ \tau^{N}_{k_{\varepsilon}}>T\right)$
$\displaystyle\leq\varepsilon,$

since the last summand equals to zero by construction of the set
$K_{p}(\varepsilon)$. This finishes the proof of compact containment
condition. ∎

So far, we have proved the sequence $\\{\xi^{N}\\}$ 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 $\\{\xi^{N}\\}$ has a further subsequence converging weakly to
some process $\xi$ on $M$.

We close this section by showing any limit process of $\\{\xi^{N}\\}$ has
continuous paths almost surely.

######  Proposition 4.10.

Any limit point $\xi$ of geodesic random walks $\\{\xi^{N}\\}$ is a.s.
continuous.

###### Proof.

The uniform elliptic condition implies that the jump sizes of the geodesic
random walks $\xi^{N}$ converge to zero uniformly as $N\to\infty$, since

$\displaystyle d(\xi^{N}_{t-},\xi^{N}_{t})\leq\frac{C+1}{\sqrt{N}},\ \forall
t\in[0,\infty).$ (4.66)

Hence the statement follows immediately from Theorem 3.10.2 of [14]. ∎

###  4.4 Convergence of geodesic random walks.

In this section, we give the proof of Theorem 2.1. We already know the
sequence $\\{\xi^{N}\\}$ is relatively compact. To show the weak convergence,
it remains to prove all limit points of $\\{\xi^{N}\\}$ 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 $A_{N}$ .

######  Lemma 4.11.

For any $p\in M$, any limit point $\xi$ of $\\{\xi^{N}\\}$ and any
$f\in\mathcal{C}^{\infty}_{K}$, we have

$\displaystyle
f(\xi_{t})-f(p)-\int_{0}^{t}Af(\xi_{s})\,\mathrm{d}s,\quad\forall t\geq 0,$
(4.67)

is a martingale.

###### Proof.

It suffices to show that for any $l\geq 1$, any
$h_{1},\dots,h_{l}\in\mathcal{C}_{b}(M)$, any $0\leq s\leq t$,
$s_{1},\dots,s_{l}\in[s,t]$, and any $f\in\mathcal{C}^{\infty}_{K}$, the
following holds.

$\displaystyle\mathbf{E}\Big{[}\Big{(}f(\xi_{t})-f(\xi_{s})-\int_{s}^{t}Af(\xi_{r})\,\mathrm{d}r\Big{)}\prod_{j=1}^{l}h_{j}(\xi_{s_{j}})\Big{]}=0.$
(4.68)

Since $\xi^{N}$ is a Markov process for all $N\geq 1$, it follows for all
$0\leq s\leq t$ that

$\displaystyle
f(\xi^{N}_{t})-f(\xi^{N}_{s})-\int^{t}_{s}A_{N}f(\xi^{N}_{r})\,\mathrm{d}r$

is a martingale. Hence for each $N\geq 1$

$\displaystyle\mathbf{E}\Big{[}\Big{(}f(\xi^{N}_{t})-f(\xi^{N}_{s})-\int_{s}^{t}A_{N}f(\xi^{N}_{r})\,\mathrm{d}r\Big{)}\prod_{j=1}^{l}h_{j}(\xi^{N}_{s_{j}})\Big{]}=0.$
(4.69)

Separate the formula (4.68) into two terms, and let $\\{\xi^{N_{k}}\\}$ be a
subsequence converging weakly to $\xi$. Since $\xi$ has continuous paths
almost surely, the finite dimensional distributions of $\xi^{N_{k}}$ always
converge weakly to those of $\xi$ (Theorem 3.7.8 of [14]). Thus we have

$\displaystyle\mathbf{E}\Big{[}\Big{(}f(\xi_{t})-f(\xi_{s})\Big{)}\prod_{j=1}^{l}h_{j}(\xi_{s_{j}})\Big{]}=\lim_{N_{k}\to\infty}\mathbf{E}\Big{[}\Big{(}f(\xi^{N_{k}}_{t})-f(\xi^{N_{k}}_{s})\Big{)}\prod_{j=1}^{l}h_{j}(\xi^{N_{k}}_{s_{j}})\Big{]}.$
(4.70)

Furthermore,

$\displaystyle\mathbf{E}\Big{[}\int_{s}^{t}A_{N_{k}}f(\xi^{N_{k}}_{r})\,\mathrm{d}r\cdot\prod_{j=1}^{l}h_{j}\left(\xi^{N_{k}}_{s_{j}}\right)\Big{]}$
$\displaystyle=\mathbf{E}\Big{[}\int_{s}^{t}Af(\xi^{N_{k}}_{r})\,\mathrm{d}r\cdot\prod_{j=1}^{l}h_{j}\left(\xi^{N_{k}}_{s_{j}}\right)\Big{]}$
(4.71)
$\displaystyle+\mathbf{E}\Big{[}\int_{s}^{t}(A_{N_{k}}-A)f(\xi^{N_{k}}_{r})\,\mathrm{d}r\cdot\prod_{j=1}^{l}h_{j}\left(\xi^{N_{k}}_{s_{j}}\right)\Big{]}$

and the latter summand vanishes as $N_{k}\to\infty$ because the functions
$h_{j}$ are bounded and by Proposition 4.3

$\displaystyle\lim_{N\rightarrow\infty}\lVert(A_{N_{k}}-A)f\rVert=0,\quad\forall
f\in\mathcal{C}^{\infty}_{K}.$

To treat the first term, since $x\mapsto Af(x)$ is continuous and bounded, we
have for each $r\in[s,t]$

$\displaystyle\lim_{N_{k}\rightarrow\infty}\mathbf{E}\Big{[}Af(\xi^{N_{k}}_{r})\cdot\prod_{j=1}^{l}h_{j}\left(\xi^{N_{k}}_{s_{j}}\right)\Big{]}=\mathbf{E}\Big{[}Af(\xi_{r})\cdot\prod_{j=1}^{l}h_{j}(\xi_{s_{j}})\Big{]}.$
(4.72)

Thus by Fubini’s and Lebesgue’s theorems we get

$\displaystyle\lim_{N\to\infty}\mathbf{E}\Big{[}\int_{s}^{t}Af(\xi^{N}_{r})\,\mathrm{d}r\cdot\prod_{j=1}^{l}h_{j}(\xi^{N}_{s_{j}})\Big{]}$
$\displaystyle=\int_{s}^{t}\lim_{N\to\infty}\mathbf{E}\Big{[}Af(\xi^{N}_{r})\cdot\prod_{j=1}^{l}h_{j}(\xi^{N}_{s_{j}})\Big{]}\,\mathrm{d}r$
(4.73)
$\displaystyle=\mathbf{E}\Big{[}\int_{s}^{t}Af(\xi_{r})\,\mathrm{d}r\cdot\prod_{j=1}^{l}h_{j}(\xi_{s_{j}})\Big{]}.$

and (4.68) is established. ∎

######  Proposition 4.12.

The martingale problem (4.67) has a unique solution which is stochastically
complete. Hence the sequence $\\{\xi^{N}\\}$ converges weakly.

###### Proof.

The well-posedness follows from the well-posedness of the martingale problem
in $\mathbb{R}^{m}$. 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 $\\{\xi^{N}\\}$ converges weakly to
some process $\xi$ 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.

######  Proposition 4.13.

The limit process $\xi$ is Feller, i.e. its semigroup preserves
$\mathcal{C}_{0}(M)$.

###### Proof.

In any chart, $\xi$ is a non-degenerate diffusion with smooth coefficients,
hence its semigroup maps $\mathcal{C}_{0}(M)$ to $\mathcal{C}(M)$.

Denote $(T_{t})_{t\geq 0}$ the semigroup of $\xi$ as usual. Let
$f\in\mathcal{C}_{0}(M)$, $t>0$ and $\varepsilon>0$ be fixed. Choose a compact
set $C_{\varepsilon}$ such that $|f(x)|\leq\varepsilon$ for $x\notin
C_{\varepsilon}$. Define $R(\varepsilon,t)$ as in (4.64) in Lemma 4.9. By this
lemma, for any $p\in M$ such that $d(p,C_{\varepsilon})>R(\varepsilon,t)$, we
have

$\displaystyle|\mathbf{E}_{p}f(\xi_{t})|$
$\displaystyle\leq\mathbf{E}_{p}|f(\xi_{t})|\mathbb{I}(\tau^{R}\leq
t)+\mathbf{E}_{p}|f(\xi_{t})|\mathbb{I}(\tau^{R}>t)$ (4.74)
$\displaystyle\leq\|f\|\cdot\mathbf{P}_{p}(\tau^{R}\leq
t)+\varepsilon\leq(\|f\|+1)\varepsilon.$

Thus $(T_{t})(f)$ vanishes at infinity for $f\in\mathcal{C}_{0}$. As $\xi$ is
a limit point of $\\{\xi^{N}\\}$, Lemma 4.7 implies

$\displaystyle\sup_{p\in M}\mathbf{P}_{p}(d(p,\xi_{t})>\delta)\leq
C(\delta)t,\ \forall\delta>0,\ \forall t\geq 0.$

The strong continuity of the semigroup $(T_{t})$ follows. ∎

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Tianyu Ma |
---|---
Institut für Mathematik |
Friedrich–Schiller Universität Jena |
07737 Jena, Germany |
<EMAIL_ADDRESS>|

Vladimir S. Matveev |
---|---
Institut für Mathematik |
Friedrich–Schiller Universität Jena |
07737 Jena, Germany |
<EMAIL_ADDRESS>|

Ilya Pavlyukevich |
---|---
Institut für Mathematik |
Friedrich–Schiller Universität Jena |
07737 Jena, Germany |
<EMAIL_ADDRESS>|