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 $X$.
Best Answer
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.