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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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  • $\begingroup$ 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. $\endgroup$
    – Nate Eldredge
    Commented Jul 22, 2013 at 21:20
  • $\begingroup$ hi, Nate Eldredge. I am afraid Donsker's theorem is nothing to do with the triangulation. $\endgroup$
    – shu
    Commented Jul 22, 2013 at 22:01
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    $\begingroup$ 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? $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 23, 2013 at 3:15
  • $\begingroup$ 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? $\endgroup$
    – Noah Stein
    Commented Jul 23, 2013 at 15:44
  • $\begingroup$ @Noah Stein, but the limit process might be a universal objet... $\endgroup$
    – shu
    Commented Jul 23, 2013 at 15:53

1 Answer 1

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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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  • $\begingroup$ 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. $\endgroup$
    – shu
    Commented Jul 23, 2013 at 21:03
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    $\begingroup$ I guess "good approximation" means that an error of some sort goes to zero when the mesh of the discretization goes to zero? $\endgroup$
    – Did
    Commented Jul 24, 2013 at 7:43

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