# Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold

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## Abstract

We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold.

## Keywords

Symmetric exclusion process Compact Riemannian manifold Hydrodynamic limit Random grids## 1 Introduction

Hydrodynamic limits of interacting particle systems is a well established subject. A large variety of parabolic equations (such as the non-linear heat equation) and hyperbolic conservation laws have been obtained from microscopic stochastic particle systems; see DeMasi and Presutti [7], Kipnis and Landim [13], Seppäläinen [17] for overviews. Usually, the setting here is that in the underlying particle system the particles move on the lattice \(\mathbb {Z}^d\), and after rescaling the limiting partial differential equation is defined on \(\mathbb {R}^d\), or on a subdomain of \(\mathbb {R}^d\) such as an interval, where then equations with boundary conditions on the ends of the interval are derived (e.g. Dirichlet boundary conditions for the case where at the right and left end the system is coupled to a reservoir fixing the density of particles, see Gonçalves [10]).

Motivated e.g. by the study of the motion of proteins in a cell-membrane, or more general motion of particles on curved interfaces, it is clear that there are many relevant physical systems of which the macroscopic motion takes place on a Riemannian manifold rather than on Euclidean space. It is the aim of this paper to provide first steps in this direction, by considering the simplest interacting particle system on a suitable discretization of a Riemannian manifold and proving its hydrodynamic limit. The symmetric exclusion process is a well-known and well-studied interacting particle system for which in standard setting it is rather straightforward to obtain the hydrodynamic limit using duality. Duality allows to translate the one-particle scaling limit, i.e., the fact that the rescaled single particle position converges to Brownian motion to the fact that the hydrodynamic limit of the particle system is the diffusion equation. Another manifestation of duality is the fact that the microscopic equation for the expectation of the density field is already a closed equation. We consider the symmetric exclusion process on a suitable discretization (a notion defined more precisely below) of a compact Riemannian manifold and prove that its empirical density field, after appropriate rescaling, converges to the solution of the heat equation on the manifold. To obtain this result, we start in Sect. 2 by studying the invariance principle of a class of geodesic random walks, thereby extending earlier results of Jørgensen [12]. These random walks are shown to converge to Brownian motion, via the technique of generator convergence. Next, in Sect. 3, we define a notion of “uniformly approximating grids” and show that choosing uniformly *N* points on the manifold, and connecting them via a kernel depending on the Riemannian distance yields a weighted graph such that the corresponding random walk converges (as the number of random points tends to infinity) to a geodesic random walk which in turn scales to Brownian motion. We also formulate abstract conditions on approximating grids ensuring the convergence of the weighted random walk to Brownian motion. In particular, convergence of the empirical distribution to the normalized Riemannian volume in Kantorovic distance is shown to be sufficient, i.e. we show that in that setting weights can be chosen such that the corresponding random walk converges to Brownian motion. We give several examples of such suitable grids. Finally, in Sect. 4, we define the exclusion process on such suitable grids (defined in Sect. 3) and show that its empirical density converges to the solution of the heat equation, following the proof from Seppäläinen [17].

## 2 The Invariance Principle for a Class of Geodesic Random Walks

Let *M* be an *n*-dimensional, compact and connected Riemannian manifold. Then we know that *M* is complete and hence geodesically complete. The main purpose of this section is to define the geodesic random walk and to show that it approximates Brownian motion when appropriately rescaled (in time and space). Such random walks and this so-called invariance principle have been studied before (Jørgensen [12] and in a special case Blum [4]). However we will directly obtain results that are tailor-made to apply them in Sect. 3. In particular, we will obtain general assumptions on the jumping distributions of the geodesic random walk for it to converge to Brownian motion. In Sect. 2.1, we define the geodesic random walk and show convergence of the generators to the generator of Brownian motion under certain assumptions on the jumping distributions. Section 2.2 is devoted to finding out which distributions satisfy these assumptions.

### 2.1 Convergence of the Generators

**The process**

*p*with tangent vector \(\xi \) at

*p*by \(p(\cdot ,\xi )\). We denote the corresponding semigroup by

*C*(

*M*) as their domain.

*p*, it chooses a random direction \(\eta \) from \(T_pM\) with rates given by \(\mu _p\) (i.e. it waits for an exponential time with rate \(\mu _p(T_pM)\) and then independently picks a vector according to the probability distribution \(\frac{\mu _p}{\mu _p(T_pM)}\)). Then the process jumps to the position \(p(1/N,\eta )\) that is reached by following the geodesic through

*p*in the direction of \(\eta \) for time \(\frac{1}{N}\). This situation is sketched in Fig. 1. We assume that choosing random directions happens independently. In this section we will specify restrictions that the measures \(\mu _p\) should satisfy. Later (in Sect. 2.2), we will show that we can take \(\mu _p\) to be for instance the uniform distribution on the unit tangent vectors at

*p*.

**The** Open image in new window **case**

Before we go into the general case, we illustrate the above in \(\mathbb {R}^n\). In \(\mathbb {R}^n\) the exponential map is simply addition if we identify \(T_p\mathbb {R}^n\) with \(\mathbb {R}^n\) itself. So in that case from a point *p* the process moves to \(p(1/N,\eta )=p+\frac{1}{N}\eta \) where \(\eta \) is chosen from \(T_p\mathbb {R}^n=\mathbb {R}^n\) randomly. This means that the discrete time jumping process when jumping as described above, can be denoted by \(S^N_m=\sum _{i=1}^m\frac{1}{N}\eta _i=\frac{1}{N}\sum _{i=1}^m\eta _i\) where \(\eta _j\) is drawn from \(T_{S_{j-1}}\mathbb {R}^n=\mathbb {R}^n\) according to some distribution. Now let \(\{N_t,t\ge 0\}\) be a Poisson process with rate one and define \(X^N_t=S_{N_t}\). Then *X* makes the same jumps as *S*, but after independent exponential times. We see that \(X^N=\{X^N_t,t\ge 0\}\) satisfies the description above. Now the invariance principle tells us that under some conditions on the jumping rates \(X^N_{tN^2}\rightarrow B_t\) in distribution as *N* goes to infinity, where *B* is Brownian motion. We show the analogous result in the more general setting of a manifold.

**Aim**

We denote the Laplace–Beltrami operator on the manifold by \(\Delta _M\). The rest of this section will be devoted to the proof of the following result.

### Proposition 2.1

\(\sup _{p\in M} \sup _{\eta \in \mathrm {supp}\mu _p} ||\eta ||<\infty \)

\(\sup _{p\in M} \mu _p(T_pM)<\infty \)

\(\int \eta ^i \mu _p(\mathrm {d}\eta )=0\) and \(\int \eta ^i\eta ^j\mu _p(\mathrm {d}\eta )=g^{ij}(p)\) in each coordinate system around

*p*

*M*.

The first assumption requires that the supports of the measures and their total masses are bounded uniformly over all points of the manifold. We will loosely say that the measures are uniformly compactly supported and uniformly finite. Since \(C^\infty (M)\) is a core for \(\frac{1}{2}\Delta _M\) [20], the Trotter-Kurtz theorem (see Kurtz [14]) implies the following corollary.

### Corollary 2.2

In the situation of Proposition 2.1 the geodesic random walk converges to Brownian motion in distribution in \(D([0,\infty ),M)\) (the space of cadlag maps \([0,\infty )\rightarrow M\)).

Note that if we denote the random variable corresponding to \(\mu _p\) by \(\zeta _p\), the second requirement of Proposition 2.1 is that (in any coordinate system) \(\mathbb {E}\zeta _p^i = 0\) and \(\mathrm {Cov}(\zeta _p^i,\zeta _p^j)=g^{ij}(p)\). This shows that the mean vector *m* of \(\zeta _p\) satisfies \(m=0\) and the covariance matrix \(\Sigma \) satisfies \(\Sigma =(g^{ij})(p)\). In \(\mathbb {R}^n\), this simplifies to \(\mathbb {E}\zeta _p^i = 0\) and \(\mathrm {Cov}(\zeta _p^i,\zeta _p^j)=\delta ^i_j\). This is satisfied for instance when \(\mu _p\) is the uniform distribution on the sphere with radius \(\sqrt{N}\) in \(\mathbb {R}^n\). Section 2.2 deals with the question which measures satisfy the restrictions above. Some examples will be given at the end of that section as well.

### Remark 2.3

Although we study the jumping distributions later, something that can already be seen now, is that we do not require any relation between jumping measures at different points of the manifold (apart from the uniform bounds on the support and the total mass). This means that our result does not require the jumping measures to be identically distributed, so it really generalizes [12].

**Choosing Suitable Charts**

Let *f* be a fixed smooth function from now on. Since we want the convergence \(N^2L_N f\rightarrow \frac{1}{2} \Delta _Mf\) to be uniform on *M*, we cannot just consider this problem pointwise. To deal with this, we will choose specific coordinate charts.

Let \(\rho \) denote the original metric of the manifold and let *d* denote the metric that is induced by the Riemannian metric. Recall that these metrics induce the same topology. This means that we do not cause confusion when we speak about open and closed sets, continuous maps and compactness without explicitly mentioning the metric. For each \(p\in M\), let \((x_p,U_p)\) be a coordinate chart for *M* around *p*. \(U_p\) is open with respect to \(\rho \) and hence with respect to *d*. This means that there is some \(\epsilon _p>0\) such that \(G_p:=\overline{B_d(p,\epsilon _p)}\subset U_p\). Now define \(O_p=B_d(p,\epsilon /2)\). Since *M* is compact, we can find \(p_1,\ldots ,p_m\) such that \(M\subset \cup _i O_{p_i}\). We have the following easy statement.

### Lemma 2.4

Let \((g_k)_{k=1}^\infty \) and *g* be functions \(M\rightarrow \mathbb {R}\). If \(g_k\rightarrow g\) uniformly on each \(O_{p_i}\), then \(g_k\rightarrow g\) uniformly on *M*.

### Proof

Let \(\epsilon >0\). For each *i* there is an \(N_i\in \mathbb {N}\) such that for all \(k\ge N_i: \sup _{O_{p_i}}|g_k(q)-g(q)|<\epsilon \). Set \(N=\max _{1\le i\le m}N_i\) and let \(q\in M\). Then there is a *j* such that \(q\in O_{p_j}\). Now for all \(k\ge N\), we see \(k\ge N_j\), so \(|g_k(q)-g(q)|\le \sup _{O_{p_i}}|g_k(s)-g(s)|<\epsilon \). This shows that \(\sup _M |g_k(q)-g(q)|\le \epsilon \). Hence \(g_k\rightarrow g\) uniformly on *M*. \(\square \)

Now let \(j\in \{1,\ldots ,m\}\) be fixed. Call \(O:=O_{p_j}\), \(\epsilon :=\epsilon _{p_j}\), \(x:=x_{p_j}\), \(G:=G_{p_j}\) and \(U:=U_{p_j}\) (this situation is shown in Fig. 2). Because of the lemma, it suffices to show that \(N^2L_N f\rightarrow \frac{1}{2} \Delta _Mf\) uniformly on *O*.

**Technical Considerations**

To obtain good estimations later, we will need that \(p(s,\eta )\) is still in our coordinate system (*x*, *U*) and even in the set *G* when \(|s|\le \frac{1}{N}\) for *N* large enough. Since the convergence must be uniform, how large *N* must be can not depend on the point *p*. The following lemma tells us how to choose such *N*.

### Lemma 2.5

### Proof

Let \(N\ge N_\epsilon \) and let \(p\in O\). The situation of the proof is visually represented in Fig. 2. Fix \(s\in (-\frac{1}{N},\frac{1}{N})\). Without loss of generality assume \(s>0\). Note that the speed of the geodesic \(p(\cdot ,\eta )\) equals \(||\eta ||\), so at time *s*, it has traveled a distance \(s||\eta ||\) from *p*. This means that there is a path of length \(s||\eta ||\) from \(p(s,\eta )\) to *p*, so \(d(p(s,\eta ),p)\le s||\eta ||\le \frac{1}{N}K \le \frac{1}{N_\epsilon }K<\epsilon /2\). Since \(p\in O\), we know \(d(p,p_j)<\epsilon /2\). Now the triangle inequality shows that \(d(p_j,p(s,\eta ))\le d(p_j,p)+d(p,p(s,\eta ))< \epsilon /2+\epsilon /2=\epsilon \). This implies that \(p(s,\eta )\in B_d(p_j,\epsilon )\subset G\). \(\square \)

*N*larger than \(N_\epsilon \).

**Taylor Expansion**

*t*to \(p(t,\eta )\). We can locally write \(f\circ p^\eta = (f\circ x^{-1})\circ (x \circ p^\eta )\), which is a composition of smooth maps. This means that \(f\circ p^\eta \) is just a smooth map \(\mathbb {R}\rightarrow \mathbb {R}\), so we can use a Taylor expansion and obtain

*N*, \(\eta \) and

*p*. This gives us

**The First Term**

*O*is contained in a coordinate chart (

*x*,

*U*). Since \(N\ge N_\epsilon \), Lemma 2.5 guarantees us that \(p(s,\eta )\) stays in the coordinate chart for \(|s|<\frac{1}{N}\). Writing \(\eta =\sum _{i=1}^n \eta ^i \frac{\partial }{\partial x^i}|_p\), we see for \(|s|<\frac{1}{N}\):

*x*is just \(\eta ^i\)). Now the first term of (1) becomes:

**The Second Term**

*f*, this can be done since

*f*is smooth. Now we want the term above to equal

*p*and for all

*i*,

*j*: \(\int _{M_p}\eta ^i\eta ^j\mu _p(\mathrm {d}\eta )=g^{ij}(p)\).

**The Rest Term**

*O*, we have the result. Let

*N*still be larger then \(N_\epsilon \).

*N*in front of the equation, we only need to know that the rest is uniformly bounded to obtain uniform convergence. It thus suffices to show that \(\frac{\mathrm {d}^3 (f\circ p^\eta )}{\mathrm {d}t^3}(t)\) is bounded as a function of \(\eta \) with \(||\eta ||<K\) and \(t\in [0,1/N_\epsilon ]\). Lemma 2.5 shows that \(p(t,\eta )\) stays in

*G*for all such \(\eta \) and

*t*. We will use this fact multiple times.

*p*and \(\eta \)), we are done.

**Bounding**\(f_i\), \(f_{ij}\)**and**\(f_{ijk}\)

First of all, note that *f* is a smooth function on *U*. Further, \(\partial _i\) defines smooth vector field on *U*. Since \(f_i=\frac{\partial f}{\partial x^i}\) is obtained by applying \(\partial _i\) on *U* to *f*, it is a smooth function on *U*. Continuing in this way, we see that \(f_{ij}\) and \(f_{ijk}\) are also smooth functions on *U*. In particular, they are smooth functions on *G* (since it is a subset of *U*). *G* is a closed subset of the compact *M* and is hence compact itself. This implies that \(f_i\), \(f_{ij}\) and \(f_{ijk}\) are (for each choice of *i*, *j*, *k*) bounded on *G*. Since we evaluate these functions in the points \(p(s,\eta )\) for \(0\le s\le 1/N\), \(N\ge N_\epsilon \) and \(||\mu ||\le K\), our discussion above shows that we only evaluate them in points of *G*. This means that we have found bounds for \(f_i\), \(f_{ij}\) and \(f_{ijk}\).

**Bounding** \(p^i_1\)

We start with a technical lemma.

### Lemma 2.6

Let \(q\in M\) and let (*y*, *V*) be a coordinate chart around *q*. Let \(v\in T_qM\) and write \(v=v^i\partial _i\). Then \(|v^i|\le \sqrt{g^{ii}(q)}||v||\).

### Proof

*q*:

Now we can use this to show the following.

### Lemma 2.7

\(|p^i_1(t)|=\left| \frac{\mathrm {d}(x^i\circ p^\eta )}{\mathrm {d}t}(t)\right| \le \sqrt{g^{ii}(p(t,\eta ))}||\eta ||\).

### Proof

*x*,

*U*) of the tangent vector to \(p^\eta \) at time

*t*so at the point \(p(t,\eta )\in M\). Using Lemma 2.6, we see

*p*is \(||\eta ||\), so this must be its speed anywhere else along the trajectory. Hence \(||\frac{\mathrm {d}p^\eta }{\mathrm {d}t}||=||\eta ||\). Inserting this in (3) yields the result. \(\square \)

We can now easily obtain a bound for \(p^i_1\). For \(0\le t\le 1/N\) and \(||\eta ||\le K\), we know \(p(t,\eta )\) stays in *G*. \(g^{ii}\) is a smooth and hence continous function on *U*, so it is bounded on *G* (since *G* is compact). This means that \(\sqrt{g^{ii}(p(t,\eta ))}\) is bounded by some \(K^{i}\) for \(||\eta ||\le K\) and \(0\le t\le 1/N\). Now we see \(|p^i_1|\le \sqrt{g^{ii}(p(t,\eta ))}\left| \left| \frac{\mathrm {d}p^\eta }{\mathrm {d}t}\right| \right| \le K^{i}K\).

**Bounding** \(\Gamma ^i_{rs}\) **and** \(\frac{\mathrm {d}}{\mathrm {d}t}\Gamma ^i_{rs}\)

Each \(g_{ij}\) is a smooth function on *U*. This means that \(\frac{\partial g_{ij}}{\partial x^k}\) is a smooth function on *U*. This implies that \(\Gamma ^i_{rs}\) is just combination of products and sums of smooth functions, so it is smooth itself. Now, as before, \(\Gamma ^i_{rs}\) is bounded on *G*. Since we only evaluate it in \(p(t,\eta )\) with \(0\le t\le 1/N\) and \(||\eta ||\le K\), we only evaluate it in *G*, so we have bounded \(\Gamma ^i_{rs}\).

### 2.2 Stepping Distribution

**Constraints for a Stepping Distribution**

*p*:

*c*. We will look into this generalized situation and at the end we will see how to determine

*c*.

**Independence of** (5) **of Coordinate Systems**

The following lemma shows that if (5) holds for a single coordinate system, it holds for any coordinate system.

### Lemma 2.8

If (5) holds for some \(c>0\) and for some coordinate system (*x*, *U*) around *p*, then it holds for the same *c* for all coordinate systems around *p*.

### Proof

*x*,

*U*) be a coordinate system around

*p*for which (5) holds with \(c>0\) and let (

*y*,

*V*) be any other coordinate system around

*p*. It suffices to show that (5) holds with the same

*c*for

*y*. Denote the metric matrix with respect to

*x*by

*g*and the one with respect to

*y*by \({\hat{g}}\). For any \(\eta \in T_pM\) define \(\eta ^1,\ldots ,\eta ^n\) as the coefficients of \(\eta \) with respect to

*x*, so such that \(\eta =\sum _i \eta ^i \frac{\partial }{\partial x^i}\). Analogously let \({\hat{\eta }}^1,\ldots ,{\hat{\eta }}^n\) be such that \(\eta =\sum _i {\hat{\eta }}^i \frac{\partial }{\partial y^i}\). Let \(J=\frac{\partial (x^1,\ldots ,x^n)}{\partial (y^1,\ldots ,y^n)}\). If \(\eta \in T_pM\), then

*j*

*i*: \(\int \eta ^i \mu (\mathrm {d}\eta ) = 0\). Moreover, for any

*i*,

*j*: \(\int \eta ^i\eta ^j\mu (\mathrm {d}\eta )=c g^{ij}\), so for any

*i*,

*j*:

*y*with the same

*c*. \(\square \)

**Orthogonal Transformations and Canonical Measures**

We now introduce a class of measures.

### Definition 2.9

Let *V* be an inner product space and let *T* be a linear map \(V\rightarrow V\). We call *T* an *orthogonal transformation* if for any \(u,v\in V\): \(\left<Tu,Tv\right>=\left<u,v\right>\).

*canonical*if for any orthogonal transformation

*T*on \(T_pM\) and for any coordinate system:

### Remark 2.10

In the same way as above, one can show that \(\mu \) has the property above with respect to some coordinate system if and only if it has the property with respect to every coordinate system. Moreover, since \(-I\) always satisfies \((-I)^TG(-I)=G\), we see that \(\int \eta ^i \mu (\mathrm {d}\eta )=\int (-\eta )^i \mu (\mathrm {d}\eta )=\int -\eta ^i \mu (\mathrm {d}\eta )=-\int \eta ^i \mu (\mathrm {d}\eta )\), so \(\int \eta ^i \mu (\mathrm {d}\eta )\) is 0 for any canonical \(\mu \).

In words, \(\mu \) is canonical if orthogonal transformations do not change the mean vector and the covariance matrix of a random variable that has distribution \(\mu \). Remark 2.10 shows that in fact the mean vector must be 0. Note that in particular measures that are invariant under orthogonal transformations are canonical, since then \(\int (T\eta )^i \mu (\mathrm {d}\eta ) = \int \eta ^i (\mu \circ T^{-1})(\mathrm {d}\eta ) = \int \eta ^i\mu (\mathrm {d}\eta )\) and the other equation follows analogously. However a simple example shows that the converse is not true. Let \(M=\mathbb {R}\) and let \(\mu \) be any non-symmetric distribution on \(T_pM=\mathbb {R}\) with mean 0. The only orthogonal transformation (apart from the identity) is \(t\mapsto -t\). Under this transformation the mean (which is 0) and the second moment are obviously left invariant, but \(\mu \) is not symmetric, so it is not invariant. We will give an example for \(\mathbb {R}^n\) later.

*x*,

*U*) is some coordinate system around

*p*and \(G=(g_{ij})\) is the matrix of the metric in

*p*with respect to

*x*, we can write a linear transformation \(T:T_pM\rightarrow T_pM\) as a matrix (which we will also call

*T*) with respect to the base \(\frac{\partial }{\partial x^1},\ldots ,\frac{\partial }{\partial x^n}\). We see that

*T*is orthogonal, this must equal

Now for a measure \(\mu \) on \(T_pM\) and a coordinate system (*x*, *U*), define the vector \(A_\mu \) and the matrix \(B_\mu \) by \(A_\mu ^i=\int \eta ^i\mu (\mathrm {d}\eta )\) and \(B_\mu ^{ij}=\int \eta ^i\eta ^j\mu (\mathrm {d}\eta )\). Then we have the following.

### Lemma 2.11

- (i)
\(\mu \) is canonical.

- (ii)
For every linear transformation

*T*and every coordinate system (*x*,*U*): if \(G=T^TGT\) , then \(A_\mu =TA_\mu \) and \(B_{\mu }=TB_\mu T^T\).

### Proof

*ii*) is just the definition of being canonical written in local coordinates. Indeed, we already saw that orthogonality or

*T*translates in local coordinates to \(G=T^TGT\), the other expressions follow in a similar way from the following equations:

**Canonical Measures are Stepping Distributions**

Now we have the following result.

### Proposition 2.12

Let \(\mu \) be a probability measure on \(T_pM\). Then \(\mu \) is canonical if and only if it satisfies (5) for some \(c>0\).

### Proof

First assume that \(\mu \) is canonical and let (*x*, *U*) be normal coordinates centered at *p*. Because of Lemma 2.8 it suffices to verify (5) for *x*, so we need to show that \(A_\mu =0\) and \(B_\mu =cG^{-1}=cI\) for some \(c>0\).

The fact that \(A_\mu =0\) is just Remark 2.10. Now note that since \(B_\mu \) is symmetric, it can be diagonalized as \(TB_\mu T^{-1}\) where *T* is an orthogonal matrix (in the usual sense). This means that \(T^T=T^{-1}\) and that \(T^TGT=T^TIT=T^TT=I=G\), so Lemma 2.11 tells us that the diagonalization equals \(TB_\mu T^T=B_\mu \). This implies that \(B_\mu \) is a diagonal matrix. Now for \(i\ne j\) let \({\bar{I}}^{ij}\) be the \(n\times n\)-identity matrix with the \(i^\text {th}\) and \(j^\text {th}\) column exchanged. It is easy to see that \(({\bar{I}}^{ij})^T{\bar{I}}^{ij}=I\), so we must also have \(B_\mu ={\bar{I}}^{ij}B_\mu ({\bar{I}}^{ij})^T\). The latter is \(B_\mu \) with the \(i^\text {th}\) and \(j^\text {th}\) diagonal element exchanged. This shows that these elements must be equal. Hence all diagonal elements are equal and \(B_\mu =cI\) for some \(c\in \mathbb {R}\). Since \(c=B_\mu ^{11}=\int \eta ^1\eta ^1\mu (\mathrm {d}\eta )\ge 0\), we know that \(c\ge 0\). If \(c=0\), then \(B_\mu =0\), so \(\mu =0\), which is not possible. We conclude that \(c>0\).

Conversely let (*x*, *U*) be a coordinate system with corresponding metric matrix *G* and assume that \(\mu \) satisfies (5) for some \(c>0\). Let *T* be such that \(G=T^TGT\). Then \(A_\mu =0=T0=TA_\mu \). We also see: \(T^TGT=G \iff G=(T^T)^{-1}GT^{-1} \iff G^{-1}=TG^{-1}T^T \iff cG^{-1}=T(cG^{-1})T^T \implies B_\mu =TB_\mu T^T\) (since \(B_\mu =cG^{-1}\)), so by Lemma 2.11\(\mu \) is canonical. \(\square \)

Now we know that if the stepping distribution is canonical (and finite and compactly supported, uniformly on *M*), the generators converge to the generator of Brownian motion that is speeded up by some factor \(c>0\) (depending on \(\mu \)). The question remains what this *c* is. The following lemma answers this question.

### Lemma 2.13

Suppose \(\mu \) satisfies (5) for some \(c>0\). Then \(c=\frac{\int ||\eta ||^2\mu (\mathrm {d}\eta )}{n}\).

### Proof

*x*,

*U*)):

The nice part of this lemma is that the expression for *c* does not involve a coordinate system, only the norm (and hence inner product) of \(T_pM\). In particular we see that \(c=1\) is equivalent to \(\int ||\eta ||^2\mu (\mathrm {d}\eta )=n\). We summarize our findings in the following result.

### Proposition 2.14

A probability measure \(\mu \) on \(T_pM\) satisfies (5) for some \(c>0\) if and only if it is canonical and \(c=\frac{\int ||\eta ||^2\mu (\mathrm {d}\eta )}{n}\). In particular, it satisfies (4) if and only if it is canonical and \(\int ||\eta ||^2\mu (\mathrm {d}\eta )=n\).

### Remark 2.15

Note that all we need of the jumping distributions is that their mean is 0, their covariance matrix is invariant under orthogonal transformations, they are (uniformly) compactly supported and they are (uniformly) finite. We don’t need the measures to be similar in any other way, so we do not at all require the jumps to have identical distributions in the sense of Jørgensen [12].

### Examples

- 1.
To satisfy (4) for every coordinate system, by Lemma 2.8 it suffices to choose a coordinate system and construct a distribution that satisfies (4) for that coordinate system. Let (

*x*,*U*) be any coordinate system around some point in*M*with corresponding metric matrix*G*in that point. Let*X*be any random variable in \(\mathbb {R}^n\) that has mean vector 0 and covariance matrix \(G^{-1}\) (for instance let \(X\sim N(0,G^{-1})\)). Now let \(\mu \) be the distribution of \(\sum _i X^i\frac{\partial }{\partial x^i}\). Then by construction \(\int \eta ^i\mu (\mathrm {d}\eta )=\mathbb {E}X^i = 0\) and \(\int \eta ^i\eta ^j\mu (\mathrm {d}\eta )=\mathbb {E}X^iX^j = \mathbb {E}X^iX^j -\mathbb {E}X^i\mathbb {E}X^j=g^{ij}\). - 2.
In the previous Example (4) is immediate. Let us now consider an example that illustrates the use of Proposition 2.14. Let \(\mu _p\) be the uniform distribution on \(\sqrt{n}S_pM\) (the vectors with norm \(\sqrt{n}\)). By definition of such a distribution, it is invariant under orthogonal transformations (rotations and reflections), so it is a canonical distribution. Since also \(\int ||\eta ||^2\mu (\mathrm {d}\eta ) = \int \sqrt{n}^2 \mu (\mathrm {d}\eta )=n\), we conclude that the uniform distribution on \(\sqrt{n}S_pM\) satisfies (4). Moreover, \(\sup _{p\in M} \sup _{\eta \in \mathrm {supp}\mu _p} ||\eta ||=\sqrt{n}<\infty \) and \(\sup _{p\in M} \mu _p(T_pM)=1<\infty \). Together this shows that the \(\mu _p\)’s satisfy the assumption of proposition 2.1.

- 3.
Let us conclude by showing for \(\mathbb {R}^n\) that the class of canonical distributions is strictly larger than the class of distributions that are invariant under orthogonal transformations, even with the restriction that \(\int ||\eta ||^2\mu (\mathrm {d}\eta )=n\). It suffices to find a distribution \(\mu \) with mean 0 and covariance matrix

*I*(since then \(\mu \) satisfies (4) and 2.14 then tells us that \(\mu \) is canonical and has \(\int ||\eta ||^2\mu (\mathrm {d}\eta )=n\)) and an orthogonal*T*such that \(\mu \ne \mu \circ T^{-1}\). Let \(\nu \) be the distribution on \(\mathbb {R}\) given by \(\nu =\frac{1}{5}\delta _{-2}+\frac{4}{5}\delta _{1/2}\). Then, using the natural coordinate system, \(\int t \nu (\mathrm {d}t)=\frac{1}{5}(-2)+\frac{4}{5}\frac{1}{2}=0\) and \(\int t^2\mu (\mathrm {d}t) = \frac{1}{5}(-2)^2+\frac{4}{5}(\frac{1}{2})^2=1\). Now let \(\mu =\nu \times \cdots \times \nu \) (*n*times). Then we directly see that the mean vector is 0 and the covariance matrix is*I*. However \(T=-I\) is an orthogonal transformation and \(\mu \circ (-I)^{-1}\) equals the product of*n*times \(\frac{1}{5}\delta _{2}+\frac{4}{5}\delta _{-1/2}\), so obviously \(\mu \ne \mu \circ (-I)^{-1}\).

## 3 Uniformly Approximating Grids

We would like to consider interacting particle systems such as the symmetric exclusion process on a manifold. Because the exclusion process does not make sense directly in a continuum, we need a proper discrete grid approximation. More precisely, we need a sequence of grids on the manifold that converges to the manifold in a suitable way. It will become clear that the grids will need to approximate the manifold in a uniform way. We will see in Sect. 4 that a natural requirement on the grids is that we can define edge weights (or, equivalently, random walks) on them, such that the graph Laplacians converge to the Laplace-Beltrami operator in a suitable sense.

*M*and construct a sequence of grids \((G^N)_{N=1}^\infty \) by setting \(G^N=\{p_1,\ldots ,p_N\}\). On each \(G^N\), we would like to define a random walk \(X^N\) which jumps from \(p_i\) to \(p_j\) with (symmetric) rate \(W^N_{ij}\) with the property that there exists some function \(a:\mathbb {N}\rightarrow [0,\infty )\) and some constant \(C>0\) such that for each smooth \(\phi \)

### Definition 3.1

We call a sequence of grids and corresponding weights \((G_N,W_N)_{N=1}^\infty \) uniformly approximating grids if they satisfy (6).

### Remark 3.2

*S*be the one-dimensional torus. Let \(S^N\) be the grid that places a grid point in \(k/N, k=1,\ldots ,N\). Now we can define a nearest neighbour random walk by putting \(W^N_{ij}=\mathbb {1}_{|p_i-p_j|=1/N}\). Also set \(a(N)=N^2\). Then we see for a point \(p_i\in S^N\) for \(N=2^m\) for some \(m\in \mathbb {N}\) that

We will show in Sect. 4 that if we define the Symmetric Exclusion Process on uniformly approximating grids we can prove that its hydrodynamic limit satisfies the heat equation on *M*.

It is not obvious how uniformly approximating grids could be defined. Most natural grids in Euclidean settings involve some notion of equidistance, scaling or translation invariance. All of these concepts are very hard if not intrinsically impossible to define on a manifold. The current section is dedicated to showing that uniformly approximating grids actually exist. To be more precise, we will show that a sequence \((p_n)_{n=1}^\infty \) can be used to define such grids if the empirical measures \(1/N\sum _{i=1}^N\delta _{p_i}\) converge to the uniform distribution in Kantorovich sense. In Sect. 3.4 we will show that such sequences exist: they are obtained with probability 1 when sampling uniformly from the manifold, i.e. from the normalized Riemannian volume measure.

For the calculations of this section, we need a result that forms the core of proving the invariance principle, which we have proved in Sect. 2.

### Remark 3.3

At first sight the requirement that the empirical measures approximate the uniform measure and that the grid points can be sampled uniformly seems arbitrary, but this is actually quite natural. We want to construct a random walk with symmetric jumping rates (we need this for instance for the Symmetric Exclusion Process later). This implies that the invariant measure of the random walk is the counting measure, so the random walk spend on average the same amount of time in each point of the grid. Hence the amount of time that the random walk spends in some subset of the manifold is proportional to the amount of grid points in that subset. Since we want the random walk to approximate Brownian motion and the volume measure is invariant for Brownian motion, we want the amount of time that the random walk spends in a set to be proportional to the volume of the set. This means that the amount of grid points in a subset of *M* should be proportional to the volume of that subset. This suggests that the empirical measures \(1/N\sum _{i=1}^N\delta _{p_i}\) should in some sense approximate the uniform measure. Moreover, a natural way to let the amount of grid points in a subset be proportional to its volume is by sampling grid points from the uniform distribution on the manifold.

### 3.1 Model and Motivation

**Motivation**

In statistical data analysis the following setting is known and used in various contexts such as data clustering, dimension reduction, computer vision and statistical learning, see: Singer [18], von Luxburg et al. [22], Giné et al. [9], Belkin and Niyogi [3] and Belkin [2] and references therein for general background and various applications. Suppose we have a manifold *M* that is embedded in \(\mathbb {R}^m\) for some *m* and we would like to recover the manifold from some observations of it, say an i.i.d. sample of uniform random elements of *M*. To do this we can describe the observations as a graph with as weight on the edge between two points a semi positive kernel with bandwidth \(\epsilon \) applied to the Euclidean distance between those points. Then it can be shown that the graph Laplacian of the graph that is obtained in this way converges in a suitable sense to the Laplace-Beltrami operator on *M* as the number of observations goes to infinity and \(\epsilon \) goes to 0. This suggests that we could define random walks on such random graphs and that the corresponding generators converge to the generator of Brownian motion. We generalize this idea by taking a more general sequence of graphs, but our main example (in Sect. 3.4) will be this random graph.

The main distinction between the statistical literature and our context is the following: for our purposes it is much more natural to view the manifold *M* on its own instead of embedded in a possibly high dimensional Euclidean space. This means that we have to use the distance that is induced by the Riemannian metric instead of the Euclidean distance. The latter is more suitable to purposes in statistics, because in that setting the Riemannian metric on *M* is not known beforehand. Also, a lot is known about the behaviour of the Euclidean distance in this type of situation and not so much about the distance on the manifold. We will have to make things work in *M* itself.

The problem of discretizing the Laplacian on a manifold (without embedding in a Euclidean space) is also studied in the analysis literature where the main concern is the convergence of spectra, see for instance: Burago et al. [5], Fujiwara [8] and Aubry [1], where structures like \(\epsilon \)-nets or triangulations are used to discretize the manifold. However, since we want to define the exclusion process on our discrete weighted graph which approximates the manifold, it is important that the edge weights are symmetric. Therefore these papers cannot be applied in our context.

**Model**

*M*be a compact and connected Riemannian manifold. We call a function

*f*on

*M*Lipschitz with Lipschitz constant \(L_f\) if

*M*such that \(\mu ^N:=\frac{1}{N}\sum _{i=1}^N\delta _{p_i}\) converges in the Kantorovich sense to \({\bar{V}}\) (the uniform distribution on

*M*), i.e.

*f*on

*M*that have Lipschitz constant \(L_f\le 1\). Define the \(N^{\text {th}}\) grid \(V_N\) as \(V_N=\{p_1,\ldots ,p_N\}\). Set

*k*a kernel. Define

*d*is the Riemannian metric on

*M*. Note that the only dependence on

*N*is through \(\epsilon \), hence the notation \(W^\epsilon _{ij}\) instead of \(W^N_{ij}\). These jumping rates define a random walk on \(V_N\). If we regard to points \(p_i,p_j\) as having an edge between them if \(W^N_{ij}>0\), we want the resulting graph to be connected (to make sense of the random walk and later of the particle systems defined on it). If we assume that there is some \(\alpha \) such that \(k(x)>0\) for \(x\le \alpha \), one can show that the resulting graph is connected for

*N*large enough. The main reason is that the distance between points that are close to each other goes to zero faster than \(\epsilon \). The details of the proof are in the appendix (see also Remark 3.6). Finally we define

*N*goes to infinity (and hence the bandwidth \(\epsilon \) goes to 0)

### Remark 3.4

### Remark 3.5

*a*(

*N*) is natural, we can write

*k*is a kernel that is rescaled by \(\epsilon \) inside, we need the \(1/\epsilon ^d\) to make sure the integral of the kernel stays of order 1 as \(\epsilon \) goes to 0. Since the amount of points that the process can jump to equals

*N*, we also need the factor 1 /

*N*to make sure the jumping rate is of order 1 as

*N*goes to infinity. Also note that the typical distance that a particle jumps with these rates is of order \(\epsilon \). This means that space is scaled by \(\epsilon \). Hence it is very natural to expect that time should be rescaled by \(1/\epsilon ^2\), which is exactly what we have.

Finally note that in the calculations *N* is the main parameter and \(\epsilon \) an auxiliary parameter depending on *N*. However, conceptually, when the scaling is concerned, the most important parameter is \(\epsilon \). *N* is just the total amount of positions and simply has to grow fast enough as \(\epsilon \) goes to 0. To see why this is true, note that any sequence \(\epsilon (N)\) that goes to 0 more slowly than what we use here will also do. Hence \(\epsilon \) should go to 0 slow enough with respect to *N* or, equivalently, *N* should go to infinity fast enough with respect to \(\epsilon \).

### Remark 3.6

We mentioned earlier that *N* must grow to infinity fast enough as \(\epsilon \) goes to 0. In fact, with \(\epsilon \) as defined in (7), the number of points in a ball of radius \(\epsilon \) goes to infinity (even though \(\epsilon \) shrinks to 0). In particular, this means that the number of points that a particle can jump to, goes to infinity. This is very different from the \(\mathbb {R}^n\) case with the lattice approximation \(\frac{1}{N}\mathbb {Z}^d\), where the number of neighbours is constant. The reason why it should be different in the manifold case is the following. In \(\mathbb {R}^d\), the natural grid \(\frac{1}{N}\mathbb {Z}^d\) is very symmetric. Indeed, we can split the graph Laplacian into the contributions \(N^2(f(x+\mathrm {e}_i/N)+f(x-\mathrm {e}_i/N)-2f(x))\) in each direction *i*, where \(\mathrm {e}_i\) it the unit vector in direction *i*. Now when applying Taylor we see that the first order terms cancel perfectly, leaving us only with the second order terms, which we want for the Laplacian. In a manifold such perfect cancellation is not possible. Therefore the way to make the first order terms cancel is to sample more and more points around a grid point, such that the sum over the linear order terms becomes an integral which then vanishes in the limit. For this reason we need the number of points in a ball of size \(\epsilon \) to go to infinity.

### Remark 3.7

It is also possible to define \(W_{ij}^N\) as \(p_\epsilon (p_i,p_j)\), the heat kernel after time \(\epsilon \), and rescale by \(\epsilon ^{-1}\) instead of \(\epsilon ^{-2-d}\). Then the result of Sect. 3.2 can be proven in the same way (by obtaining some good bounds on Lipschitz constants and suprema of the heat kernel and choosing \(\epsilon =\epsilon (N)\) appropriately, see Cipriani and van Ginkel [6]) and the result of Sect. 3.3 is a direct consequence of the fact that the Laplace-Beltrami operator generates the heat semigroup. However, for purposes of application/simulation the weights that we have chosen here are much easier to calculate (since only the geodesic distances need to be known, not the heat kernel).

### 3.2 Replacing Empirical Measure by Uniform Measure

*C*independent of

*i*such that for all smooth

*f*

*k*is Lipschitz so it has some Lipschitz constant \(L_k<\infty \). This implies that

*f*is smooth, so it is Lipschitz too with Lipschitz constant \(L_f\). Since \(f(p_i)\) is just a constant, \(f(\cdot )-f(p_i)\) is also Lipschitz with Lipschitz constant \(L_f\). Since they are both bounded functions, we see for the Lipschitz constant of \({g^{\epsilon ,j}}\):

*k*is bounded since it is Lipschitz and compactly supported, so \(||k||_\infty <\infty \). This shows that:

*N*explicitly. By (7), \(W_1(\mu ^N,\nu )\le \epsilon (N)^{4+d}\), so we obtain

**What Remains**

*N*via \(\epsilon \) and \(\epsilon (N)\downarrow 0\) as \(N\rightarrow \infty \). Since the \(p_i\)’s are all in

*M*we can replace \(p_i\) by

*q*and require that the convergence is uniform in \(q\in M\).

*M*. The process jumps from

*p*to a (measurable) set \(Q\subset M\) with rate \(\int _Q \epsilon ^{-2-d}k(d(p,q)/\epsilon )\mathrm {d}{\bar{V}}\).

### Remark 3.8

*H*(

*x*) is the Hessian of

*f*in

*x*. Now changing coordinates to integrate over each sphere \(B_r\) of radius

*r*with respect to the appropriate surface measure \(S_r\) and then with respect to

*r*, we obtain

*i*, but only on

*r*. Therefore the first term vanishes and we are left with

### 3.3 Convergence Result

**Integral Over Tangent Space**

*k*is compactly supported). We denote for \(p\in M,r>0: B_d(p,r)=\{q\in M: d(p,q)\le r\}\). Then we can write

*M*with respect to the original metric \(\rho \)). For \(\epsilon \) small enough we know that \(\exp _p: T_pM\supset B_p(0,\alpha \epsilon ) \rightarrow B_d(p,\alpha \epsilon )\subset M\) is a diffeomorphism. We want to use this to write the integral above as an integral over \(B_p(0,\epsilon )\subset T_pM\):

**Determining the Measure** \(\bar{V}\circ \exp \circ \lambda _\epsilon \)

*p*, so there exists a normal coordinate system \((x,V_\epsilon )\) that is centered at

*p*. We interpret, for \(v\in \mathbb {R}^n\), \(v_p\in T_pM\) as \(\sum _i v_i \frac{\partial }{\partial x^i}\). Consequently, when we write \(A_p\) for some subset

*A*of \(\mathbb {R}^n\), we mean \(\{v_p: v\in A\}\). Since the basis \(W=\left( \frac{\partial }{\partial x^1}\ldots ,\frac{\partial }{\partial x^n}\right) \) is orthogonal in \(T_pM\), it is easy to see that \(\phi :=v_p\mapsto v\) preserves the inner product and is an isomorphism of inner product spaces. Indeed,

### Lemma 3.9

There exist \(\epsilon '>0\) and a function \(h:B_{\mathbb {R}^n}(0,\epsilon ')\rightarrow \mathbb {R}\) such that for *t* tending to \(0\,h(t)=O(||t||^2)\) and for all \(0<\epsilon <\epsilon '\): \(\bar{V}\circ \exp \circ \lambda _\epsilon =\epsilon ^n \left( \frac{1+h(\epsilon t)}{V(M)} \mathrm {d}t^1 \ldots \mathrm {d}t^n \right) \circ \phi \) on \(B_p(0,\alpha )\).

### Proof

Let \(\epsilon '\) be small enough such that the considerations above the lemma hold and let \(\epsilon <\epsilon '\). For clarity of the proof, we first separately prove the following statement.

*Claim:*\(x\circ \exp =\phi \) on \(B_{\mathbb {R}^n}(0,\alpha \epsilon )_p\).

### Proof

*p*are straight lines with respect to

*x*, so they are of the form \(x(\gamma (t))=ta+b\) with \(a,b\in \mathbb {R}^n\). For \(\eta =\sum _i \eta ^i\frac{\partial }{\partial x^i}\), the geodesic starting at

*p*with tangent vector \(\eta \) at

*p*should satisfy \(b=x(p)=0\) and \(a_i=\eta ^i\) for all

*i*, so we see \(\gamma ^k=t\eta ^k\). For \(q\in B_d(p,\alpha \epsilon )\), we see \(x^k(\exp (x(q)_p))=1*x^k(q)=x^k(q)\), so \(\exp (x(q)_p)=q\). This also shows that \(x\circ \exp (v_p)=v\) for \(v\in B_{\mathbb {R}^n}(0,\alpha \epsilon )\) (since

*x*is invertible), which gives an identification

Now we will first use the definition of integration to see what the measure is in coordinates (so it becomes a measure on a subset of \(\mathbb {R}^n\)). Then we will use the claim above: we will pull the measure on \(\mathbb {R}^n\) back to \(T_pM\) using \(\phi \).

*h*is such that \(h(x)=O(||x||^2)\). Now the measure can be written in local coordinates on \(B_{\mathbb {R}^n}(\alpha \epsilon ')\) as \((1+h(x))\mathrm {d}x^1\wedge \ldots \wedge \mathrm {d}x^n\), so the uniform measure is \(\frac{1+h(x)}{V(M)}\mathrm {d}x^1\wedge \ldots \wedge \mathrm {d}x^n\). This yields the measure \({\bar{V}}\circ x^{-1}=\frac{1+h(t)}{V(M)} \mathrm {d}t^1 \ldots \mathrm {d}t^n\) on \(x(V_{\epsilon '})=B_{\mathbb {R}^n}(0,\alpha \epsilon ')\). We have on \(B_{\mathbb {R}^n}(0,\alpha )_p\):

### Remark 3.10

*p*in normal coordinates

*x*centered around

*p*:

*M*in

*p*. This implies that the way that the uniform distribution on a ball around

*p*in

*M*is pulled back to the tangent space via the exponential map depends on the curvature of

*M*in

*p*. In particular, if there is no curvature,

*M*is locally isomorphic to a neighbourhood in \(\mathbb {R}^n\) so the same thing happens as in \(\mathbb {R}^n\). This means that we get a uniform distribution on a ball around 0 in the tangent space.

### Remark 3.11

*p*. Here

*G*(

*q*) is the metric matrix at

*q*expressed in (fixed) normal coordinates centered at

*p*. Since Open image in new window and \(\det \) are uniformly continuous in the right domains, it suffices to show that

*p*. In other words,

*C*does not depend on

*p*. For all \(p\in M\) (and for any system of normal coordinates centered at

*p*) we have the following Taylor expansion (note that for fixed \(p\,G(\exp _p(\cdot ))_{ij}\) is a map from a (subset of) \(\mathbb {R}^d\) to \(\mathbb {R}\)):

*p*, i.e. we have

*p*, we note that the functions of

*p*and

*x*appearing in the r.h.s. of (18) can be made smooth both in

*p*and

*x*. Smoothness in

*x*is obvious (within the injectivity radius) and smoothness in

*p*follows from a special choice of normal coordinates in such a way that they vary smoothly with

*p*. A choice of normal coordinates is equivalent to a choice of an orthonormal basis, so one can construct smoothly varying normal coordinates by taking a smooth section of the orthonormal frame bundle (this can only be done locally, but it is enough to have the uniformity result locally, since then by compactness one has it globally). By compactness, the injectivity radius is bounded from below by some \(\delta >0\). Now for all \(p\in M\) and \(||x||<\delta \), (18) holds and (locally) the quantities on the r.h.s. vary smoothly and therefore (again by compactness) one can show that \(C:=\sup _p C_p\) is finite.

**A Canonical Part Plus a Rest Term**

*p*in the direction of \(\eta \) for time \(\epsilon \). Now we define \(\mu ^k=k(||\cdot ||)\mu \) (so the measure which has density \(k(||\cdot ||)\) with respect to \(\mu \)) and analogously \(\mu ^k_R=k(||\cdot ||)\mu _R\). Then we can write the integral above as

### Lemma 3.12

\(\mu ^k\) is canonical. Moreover \(\int _{T_pM} ||\eta ||^2 \mu ^k(\mathrm {d}\eta )=\frac{2\pi ^{n/2}}{V(M)\Gamma (n/2)}\int _0^\infty k(r)r^{n+1}\mathrm {d}r\).

### Proof

First of all recall that *k* is continuous and compactly supported, so the integral over *k* above makes sense and is finite. Define \(\nu =\frac{1}{V(M)} \mathrm {d}t^1 \ldots \mathrm {d}t^n\) on \(B_{\mathbb {R}^n}(0,\alpha )\) and 0 everywhere else. Then we can write \(\mu =\nu \circ \phi \). Since \(\phi \) preserves the norm, we see that \(k(||\cdot ||_{T_pM})\circ \phi ^{-1}=k(||\cdot ||_{\mathbb {R}^n})\). This means that \(\mu ^k=\nu ^k\circ \phi \), where \(\nu ^k:=k(||\cdot ||)\nu \). Since \(\phi \) preserves the inner product, the measure \(\mu ^k\) behaves the same with respect to orthogonal transformations in \(T_pM\) as \(\nu ^k\) with respect to orthogonal transformations in \(\mathbb {R}^n\). Since \(\nu ^k\) is clearly preserved under such transformations, so is \(\mu ^k\). This shows that \(\mu ^k\) is canonical.

*v*|| is constant on spheres around the origin. Here \(\frac{2\pi ^{n/2}}{\Gamma (n/2)}r^{n-1}\) is the area of \(rS_{n-1}\). In the last step we used that \(\mathrm {supp}(k)\subset [0,\alpha ]\). \(\square \)

**Conclusion**

We use everything above to obtain the statement that we aim for.

### Proposition 3.13

### Proof

*p*

*p*. Let \(\epsilon '',K>0\) such that \(\epsilon ''<\epsilon '\) and \(|h(s)|<K||s||^2\) for \(s\in B_{\mathbb {R}^n}(0,\epsilon '')\) (where both \(\epsilon '\) and

*h*are from Lemma 3.9). We need Remark 3.11 to make sure that

*K*and \(\epsilon ''\) do not depend on

*p*. Now note that for \(\epsilon <\epsilon ''\):

*k*is bounded and has support in \([0,\alpha ]\). Combining everything above gives what we wanted. \(\square \)

### 3.4 Example Grid

So far, we have seen that a sequence of grids is suitable for the hydrodynamic limit problem if the empirical distributions converge to the uniform distribution in the Kantorovich topology. We conclude by giving examples of such grids. To be more precise, we show that if one constructs a grid by adding uniformly sampled points from the manifold, this grid is suitable with probability 1.

### Remark 3.14

*S*from Remark 3.2. We can show that the empirical measures corresponding to these grids along the subsequence \(N=2^m, m=0,1,2, \ldots \) converge to the uniform measure on

*S*with respect to the Kantorovich distance. To this end let \(N=2^m\) be fixed, call the corresponding empirical measure \(\mu ^N\) and call the uniform measure \(\lambda \). Recall that the Kantorovich distance between these measures is alternatively given by

*Y*be a uniform random variable on

*S*and define

*X*,

*Y*) by \(\nu \). Then it is easy to see that \(\nu \in \Gamma (\mu ^N,\lambda )\). This implies that

*not*the same as those in Remark 3.2.

**Convergence of a Random Grid**

Now we move back to the general case of a compact and connected *n*-dimensional Riemannian manifold *M*. Let \((P_n)_{n=1}^\infty \) be a sequence of iid uniformly random points of *M*. Define \( \mu ^N=\frac{1}{N}\sum _{i=1}^N \delta _{P_i}\). We follow [21, Example 5.15] to show that \(W_1(\mu ^N,{\bar{V}})\rightarrow 0\) as \(N\rightarrow \infty \). First we will show that the expectation goes to 0, then we will derive that it goes to 0 almost surely.

*N*be fixed. Let \(\mathscr {F}_1\) be the set of Lipschitz function on

*M*with Lipschitz constant \(\le 1\). Then we define for \(f\in \mathscr {F}_1\) the random variable \(X_f=\mu ^Nf-\bar{V} f\). Note that both \(\mu ^N\) and \({\bar{V}}\) are probability distributions, so \(X_f(\omega )\) is Lipschitz in

*f*for each \(\omega \):

*f*has Lipschitz constant \(\le 1\):

*M*is compact, so \(K<\infty \). Since adding constants to

*f*does not change \(X_f\), it suffices to consider \(f\in \mathscr {F}_{1,K}=\{g\in \mathscr {F}_1: 0\le g \le K\}\). It follows that for each \(f\in \mathscr {F}_{1,K}\) by writing

**Estimating the Covering Number** \(N(\mathscr {F}_{1,K},||\cdot ||_\infty ,\epsilon )\)

We now need to estimate this covering number. To do this we need an upper bound of the covering number \(N(M,d,\epsilon )\) of *M*. Since *M* is compact there exist \(a,\delta >0\) such that for all \(0<\epsilon <\delta \): \(N(M,d,\epsilon )\le a\epsilon ^{-d}\) (see for instance [16, Lemma 4.2]). Using this we can prove the following.

### Lemma 3.15

There is a \(c>0\) such that for all \(0<\epsilon <\delta \): \(N(\mathscr {F}_{1,K},||\cdot ||_\infty ,\epsilon )\le \exp {c/\epsilon ^d}.\)

### Proof

*i*be such that \(p\in V_i\). Then we see:

*Y*is an \(\epsilon \)-net for \(\mathscr {F}_{1,K}\). Hence \(N(\mathscr {F}_{1,K},||\cdot ||_\infty ,\epsilon )\le \#Y\).

All we have to do now is estimate \(\#Y\).

Now define a graph *G* with vertices \(p_1,\ldots ,p_m\) by putting an edge between \(p_i\) and \(p_j\) whenever \(d(p_i,p_j)\le \epsilon /2\). Any \(\pi ^f\) is uniquely specified by its values on the nodes of *G*. Note further that whenever we know \(\pi ^f\) for some point of the graph, there are only 3 possible values left for each of its neighbours (since neighbours are at distance at most \(\epsilon /2\)). Now \(\#Y\) is dominated by the amount of ways in which we can assign values of the type \((k+1/2)\epsilon \) to nodes of *G* while keeping this restriction into account. Define, for \(i\le 0\), \(S_i=\{p\in G: d_G(p_1,p)=i\}\), where \(d_G(p,q)\) denotes the minimum amount of edges that need to be followed to walk from *p* to *q* in *G*. Now we can start counting.

*m*is the total amount of balls as we defined at the beginning of the proof, which we chose equal to \(N(M,d,\epsilon /4)\). Now we know that for \(0<\epsilon <\delta \)

*N*large enough such that \(c_0N^{\frac{-1}{d+2}}<\delta \)). This shows that the optimal bound that we get is

*N*. This shows that

**Convergence a.s.**

### Lemma 3.16

Set (as before) \(K=\sup _{p,q\in M}d(p,q)\). Then for each \(1\le j\le N\): \(||D_jH||_\infty \le K/N\).

### Proof

*K*/

*N*apart from each other, which implies that

Now we are in position to prove the main result.

### Proposition 3.17

\(W_1(\mu ^N,{\bar{V}})\rightarrow 0\) almost surely as \(N\rightarrow \infty \).

### Proof

We conclude that sampling uniformly from the manifold yields a suitable grid with probability 1.

## 4 Hydrodynamic Limit of the SEP

In Sect. 3 we showed the existence of uniformly approximating grids. In this section we will apply such grids. We will use it to define an interacting particle system on the manifold. Then we will show that this interacting particle system has a hydrodynamic limit and that this limit satisfies the heat equation (the precise formulation is given in Theorem 4.2). We follow a standard method that is used in (Seppäläinen [17], Chap. 8) for the Euclidean case.

*M*such that \(G^N=\{p_1,\ldots ,p_N\}\). On each \(G^N\), there is a random walk \(X^N\) which jumps from \(p_i\) to \(p_j\) with (symmetric) rate \(W^N_{ij}\). We assume that there exists some function \(a:\mathbb {N}\rightarrow [0,\infty )\) and some constant \(C>0\) such that for each smooth \(\phi \)

^{1}

*a*(

*N*) by

*C*if necessary, we can assume that \(C=1\).

### Remark 4.1

Note that for the result of this section it is not necessary to construct grids from a sequence. Any sequence of finite grids such that (19) holds would do. However, since the grid that we constructed in Sect. 3 is of this form and this section partially serves as an example of the application of that grid, we formulate our results in this section in the same way.

### 4.1 Symmetric Exclusion Process

*G*. The particles are considered identical. Each particle jumps after independent exponential times with parameter 1 from

*x*to

*y*with probability

*p*(

*x*,

*y*), provided that the place that it wants to jump to is not already occupied. Otherwise, the jump is suppressed. We assume that \(p(x,y)=p(y,x)\). Let \(\eta _t\in \{0,1\}^G\) denote the configuration of the particles at time

*t*, i.e. \(\eta _t(x)=1\) if there is a particle at place \(x\in G\) at time

*t*and 0 else. We will sometimes write \(\eta (p,t)=\eta _t(p)\). For any configuration \(\eta \) and points

*x*,

*y*define \(\eta ^{xy}\) by

*xy*) have independent exponential clocks with rate \(p(x,y)=p(y,x)\). Whenever a clock rings, the particles that are at either side of the corresponding edge jump along the edge. This means that if there are no particles, nothing happens. If there is one particle, it jumps. If there are two particle, they switch places. Since we are not interested in individual particles, the configuration stays the same in the latter case. Note that in this way there can never be more than two particles at the same place. Using the notation introduced above, we see that the generator of this process is defined on the core of local functions as

**The Process**

Let \((X_i)_{i=1}^\infty \) be some sequence of (possibly degenerate) random variables taking values in \(\{0,1\}\). Set as the initial configuration \(\eta ^{N}_0(p_i)=X_i\).

### 4.2 Hydrodynamic Limit

We will use this subsection to give the basic definitions that describe the idea of a hydrodynamic limit. At a microscopic scale, the particles are just random walkers with some interaction, but at the macroscopic scale (where limits are taken in space and time), the behaviour is deterministic: it is described by a partial differential equation (in our case the heat equation).

**Path Space**

Now write *R*(*M*) for the space of Radon measures on *M* with the vague topology and let \(D=D([0,\infty ),R(M))\) denote the space of all paths \(\gamma :[0,\infty )\rightarrow R(M)\) such that \(\gamma \) is right continuous and has left limits. On this space we can define the Skohorod metric (see for instance [17, Appendix A.2.2]). Since *R*(*M*) is a Polish space, it can be shown that *D* with the Skohorod metric is a Polish space too.

**Initial Conditions and Trajectories of Particle Configurations**

*t*. In particular \(\mu _t^{N}\) is a sub-probability measure and is in

*R*(

*M*).

Instead of dealing with this problem pointwise for each *t*, we will look at trajectories. As the particles move according to the SEP, \(\gamma ^{N}:[0,\infty )\rightarrow R(M)\) defined by \(t\mapsto \mu _t^{N}\) is a random trajectory and hence a random element of *D*. It represents the positions of the particles over time. The initial configuration \(X_1,\ldots ,X_N\) and the dynamics of the SEP determine a distribution \(Q^{N}\) on *D*. In this way we obtain a sequence \((Q^{N})_{N=0}^\infty \) of measures on *D*.

**Assumption on the Initial Configuration**

*V*). We would like to show that if this initial condition is given, then at any time

*t*the configurations \(\eta _t^{N}\) have a corresponding density profile \(\rho _t\mathrm {d}{\bar{V}}\). Moreover, we want to show that \(t\mapsto \rho _t\) solves the heat equation with initial condition \(\rho _0\).

**Example of Initial Distribution**

*f*: \(\frac{1}{N}\sum _{i=1}^Nf(p_i)\rightarrow \int _Mf\mathrm {d}{\bar{V}}\).

^{2}Define the random variables \((X_i)_{i=1}^\infty \) to be independent Bernoulli random variables with \(\mathbb {E}X_i = \rho _0(p_i)\) for some continuous function \(\rho _0:M\rightarrow \mathbb {R}\) with \(0\le \rho _0\le 1\). Then we see as \(N\rightarrow \infty \):

**Main Result**

After all these definitions, we can state the main result of this section.

### Theorem 4.2

Let *M* be a complete, *n*-dimensional, connected Riemannian manifold and let \((G_N,W_N)_{N=1}^\infty \) be a sequence of uniformly approximating grids with corresponding weights. Let \(\eta ^N_t\) be particle configurations that behave according to the SEP on \((G_N,W_N)\) and let \(\mu ^N_t\) be its measure valued representation. Suppose that \(\mu _0^N\) has density profile \(\rho _0\mathrm {d}V\) for some measurable function \(\rho _0\). Then the trajectory \(t\mapsto \mu ^N_t\) converges in probability to the trajectory \(t\mapsto \rho _t\mathrm {d}V\) in the Skohorod topology, where \(t\mapsto \rho _t\) satisfies the heat equation on *M* with initial condition \(\rho _0\).

### 4.3 Convergence Result

**Dynkin Martingale**

The proof of the hydrodynamic limit follows the line of (Seppäläinen [17], Chap. 8) which is a canonical method that is also discussed in Kipnis and Landim [13]. However, in our context, there are several new technical difficulties along the way which we have to tackle. Its core calculations are based on the following Dynkin martingale result. It is a standard result and it is also proved in Seppäläinen [17]. We will formulate it in terms of our situation on a compact Riemannian manifold.

### Proposition 4.3

*L*and semigroup \(S_t\). For any function

*f*such that both

*f*and \(f^2\) are in

*D*(

*L*), define

**Application of the Proposition**

*M*. Define for \(\eta \in \{0,1\}^{G^N}\): \(f^N(\eta )=\frac{1}{N}\sum _{i=1}^N \eta (p_i)\phi (p_i)=\mu (\phi )\), where \(\mu =\frac{1}{N}\sum _{i=1}^n\delta _i\eta (p_i)\). Note that since \(L^{N}\) is the generator of a random walk on a the finite space of configurations, its domain consists of all functions on those configurations, so in particular \(f^N\) and \((f^N)^2\) are in it. Applying Theorem 4.3 in this situation shows that \(M^{N}\) defined by

**Using Convergence of the Generators**

**The Error Term**

**Convergence of the Martingale to**0

### Lemma 4.4

### Proof

*i*. This shows that

*N*goes to infinity and \(\epsilon \) goes to zero, so

**Convergence of**(27)

**to**0

**in Probability**

**Tightness of**\((Q^N)_{N=1}^\infty \)

We will need that the sequence of distributions \((Q^N)_{N=1}^\infty \) is tight. This can be shown in exactly the same way as (Kipnis and Landim [13], p.55-56). In fact all the most crucial calculations have already been performed above.

### Lemma 4.5

The sequence of distributions \((Q^N)_{N=1}^\infty \) is tight.

### Proof

*f*we can map a path \(\nu \in D([0,T],R(M))\) to the path in \(D([0,T],\mathbb {R})\) given by \(t\mapsto \nu _t(f)\). This induces a sequence of distributions \(Q^Nf^{-1}\) on \(D([0,T],\mathbb {R})\). By (Kipnis and Landim [13], Chap. 4, Prop. 1.7) and the fact that the smooth functions are uniformly dense in the set of continuous functions on a manifold, it suffices to prove the conditions of (Kipnis and Landim [13], Chapt. 4, Thm. 1.3) for {\(Q^Nf^{-1},N\ge 0\}\) for all smooth

*f*. Fix such

*f*. Since each path stays in the set of sub-probability measures, the first condition is easily satisfied. For the second condition, it suffices to prove Aldous’ tightness criterion, i.e. that

*T*. We know from equation (26) that there exists a martingale

*M*(depending on

*f*) such that

**(I)**. First of all, since \(\mu _s^N\) is a sub-probability measure and \(\Delta _M f\) is bounded:

**(II)**. Further, the calculations above show that

*K*is some positive number which exists, because of (25). This part satisfies (30) in the same way as the previous part.

**(III)**. Now for the last term, we first estimate \(\mathbb {E}\left[ (M^N_{\tau +\theta }-M^N_{\tau })^2\right] \) (as is done in (Kipnis and Landim [13], p.56)). Naturally, the expectation is taken with respect to \(Q^Nf^{-1}\). Note that because of the martingale property:

**Limit Distribution**

*D*. This implies that every one of its subsequences is also tight and therefore has a weakly convergent subsequence. If these all have the same limit, then it follows from a basic result in metric spaces that the sequence itself converges weakly to that limit. It therefore suffices for weak convergence of \((Q^N)_{N=1}^\infty \) to show that every weakly convergent subsequence of \((Q^N)_{N=1}^\infty \) has the same limit. Let \((Q^{N_k})_{k=1}^\infty \) be any weakly convergent subsequence and denote its limit by

*Q*. Since

*H*is closed, we know for any \(\delta >0\) that

**Continuity**

**Uniqueness**

To obtain uniqueness of limits of subsequences of \(Q^N\), we need to know that there is a unique continuous solution to (31) that has initial condition \(\rho _0\mathrm {d}{\bar{V}}\). We know that \(t\mapsto \rho _t\mathrm {d}{\bar{V}}\) is a continuous solution to (31) with the right initial condition if \(t\mapsto \rho _t\) satisfies the heat equation with initial condition \(\rho _0\). Therefore it suffices to show that this solution is unique. This result is proven with a boundedness condition in (Seppäläinen [17], Thm A.28). The main idea of the proof is that the measure valued path \(\alpha _t\) is smoothed by taking its convolution with some smooth kernel with bandwidth \(\epsilon >0\). Then it is shown that this trajectory of functions satisfies the heat equation with initial condition \(\rho _0\) in the strong sense (by interchanging integral and derivatives and using that these identities are known for sufficiently many \(\phi \)), so it must equal \(t\mapsto \rho _t\). Then by letting \(\epsilon \) go to zero, it is shown that the original trajectory \(t\mapsto \alpha _t\) must equal \(t\mapsto \rho _t\mathrm {d}\lambda \), where \(\lambda \) is the Lebesgue measure.

To obtain the analogous result in our setting, we cannot use convolution, since this is not well-defined on a manifold. However, we can smooth the measures by integrating the heat kernel at time \(\epsilon \) with respect to the measures. Using this smoothing, we can follow exactly the same approach, i.e. showing that the smoothed trajectory satisfies the heat equation in a strong sense and then letting \(\epsilon \) go to 0. The boundedness condition is a bound on volumes, which is needed for some estimations in Seppäläinen [17] and for the uniqueness of the strong solution to the heat equation. Since we work in a compact setting and with probability measures, such a bound is not necessary. The uniqueness of the strong solution to the heat equation is a standard result in our case (so for a compact and connected Riemannian manifold). See for instance [11, Thm. 8.18]. Results on the heat kernel on a manifold can also be found in Grigoryan [11].

**Conclusion**

Now let \(t\mapsto \rho _t\) be the solution to the heat equation on *M* with initial condition \(\rho _0\) and call \(\beta :=(t\mapsto \rho _t\mathrm {d}{\bar{V}})\). Recall that (31) holds \(Q-\)a.s. By the uniqueness result above, this implies that *Q* is a Dirac distribution with \(\beta \) as its support. Since this does not depend on \(Q^{N_k}\), it must be the same for any convergent subsequence, so with arguments given above, we conclude that \(Q^N\rightarrow Q\) weakly. Let \(\gamma ^N\) denote the random trajectory \(t\mapsto \mu ^{N}_t\). Since *Q* is degenerate, the weak convergence implies convergence in probability, so \(\gamma ^N\rightarrow \beta \) in probability. This is what we wanted to show.

## Footnotes

- 1.
Recall from Remark 3.4 that if we replace the average in this expression by a supremum, this condition implies convergence of the corresponding semigroups.

- 2.
Since Kantorovich convergence is stronger than convergence in distribution, this is in particular true for the grids that we consider in Sect. 3.

## Notes

### Acknowledgements

The authors thank Rik Versendaal for helpful discussions. The support of the grant 613.009.112 of the Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged.

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