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As we know, the random walk on $\mathbb{Z}/n$ will converge(in some sense) to the Brownian motion on $\mathbb{R}$ when $n\to\infty$. I would like to know is there some higher dimensional analogy result.

Edit: As pointed by Nate Eldredge, there is a generalization of Donsker's theorem on manifold. But, I am interested in the following more topological generalization.

For a compact Riemannian manifold $X$, if a triangulation is given. Is there a canoncal way to defined a random walk $W$ on the vertex of the triangulation, such that $W^n$ the random walk defined in this way after $n$-th barycentric subdivision will converge to the Brownian motion on $X$.

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See Reference needed: Donsker's Invariance Principle for Riemannian Manifolds; I think the references given there should answer your question. I'm tentatively marking this as duplicate. –  Nate Eldredge Jul 22 '13 at 21:20
    
hi, Nate Eldredge. I am afraid Donsker's theorem is nothing to do with the triangulation. –  shu Jul 22 '13 at 22:01
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Hi shu, Since Nate has marked this as a possible duplicate, perhaps you might expand your question to clarify why Donsker's theorem doesn't do what you want? –  Theo Johnson-Freyd Jul 23 '13 at 3:15
    
I'm not quite sure what you want out of the construction you propose. Certainly it depends on the triangulation, whereas Brownian motion doesn't, right? –  Noah Stein Jul 23 '13 at 15:44
    
@Noah Stein, but the limit process might be a universal objet... –  shu Jul 23 '13 at 15:53

1 Answer 1

Nicolas Th. Varopoulos, Brownian motion and random walks on manifolds, Annales de l'Institut Fourier 34(2) (1984), 243-269.

Abstract: We develop a procedure that allows us to “discretise” the Brownian motion on a Riemannian manifold. We construct thus a random walk that is a good approximation of the Brownian motion.

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Hi, Did. I just read the introduction. But it seems no convergence result in that paper. If I undersand correctly, they get a good approximation. And using this approximation to get some estimation on the heat kernel. –  shu Jul 23 '13 at 21:03
    
I guess "good approximation" means that an error of some sort goes to zero when the mesh of the discretization goes to zero? –  Did Jul 24 '13 at 7:43

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