In this report we study Markov processes on compact and connected Riemannian manifolds. We define a random walk on such manifolds and give a direct proof of the invariance principle. This principle says that under some conditions on the jumping distributions (i.e. the distributions of single steps), the random walk converges to Brownian motion when space is scaled by 1/N, time by N^2 and N tends to infinity (which has been shown with more general methods by Jorgensen and by Blum). To prove this, we show convergence of the generators on the set of smooth functions and we apply the Trotter-Kurtz theorem (as has been done by Blum, in a rather sketchy way and in a slightly different setting). We also show convergence of the corresponding Dirichlet forms. Then we show that the conditions on the jumping distributions are satisfied if they are compactly supported and have mean 0 and a covariance matrix which is invariant under orthogonal transformations.
Next, we define random grids on a Riemannian manifold and we define random walks on them. We show that their Dirichlet forms converge to the Dirichlet form of Brownian motion, using the results above. We also prove a result that is a bit weaker than convergence of the generators in this case.
Finally, these grids allow us to define the Symmetric Exclusion Process (SEP) on a manifold. Using the convergence results above, we follow the line of a proof of Seppäläinen to show that the hydrodynamic limit of the SEP satisfies the heat equation. Some details still need to be filled in, but we believe that this method will allow us to study interacting particle systems and their hydrodynamic limits on Riemannian manifolds.
Before all of this we start with an introduction to Markov processes, their semigroups and generators. In particular we focus on time-reversible (or symmetric) processes and the Dirichlet form with its properties. We also give an introduction to Riemannian manifolds and related notions.