Brownian motion and random walk

Question

Let $M_{\Gamma}$ a Riemannian covering of a closed compact manifold $(M,g)$ with deck transformation $\Gamma$ (its neutral element will be denoted by $e$). If we denote by $p_t^{\Gamma}(x,y)$ the heat kernel associated to $M_{\Gamma}$ does the following proposition occurs :

There exist a probability measure $\mu$ on $\Gamma$ such that the associated random walk on $\Gamma$ of law $\mu^{*n}$ satisfies that there exists two constants $C_1, C_2 > 0$ and a point $x \in M_{\Gamma}$ such that :

$$ C_2 \mu^{*n}(e,e) \le p_n^{\Gamma}(x,x) \le C_1 \mu^{*n}(e,e) $$

Answer 1

I am not an expert, so this is not a complete answer and someone maybe should comment on this.

As was stated by Did on this question, Varopoulos, in Brownian motion and random walks on manifolds, gave a discretization of the Brownian motion in the context you're referring to. I don't know if you have in there the estimates you're looking for though.

Another thing is that when $\Gamma$ is nilpotent, you can work the other way around: associate a Laplacian to a random walk. More precisely, take $\Gamma$ as a lattice in a nilpotent Lie group $N$ and consider a finitely supported probability measure $\mu$. Varapoulos and Alexopoulos gave precise asymptotics of $\mu^{*n}(x,y)$.

In particular, Alexpoulos, in the paper random walks on discrete groups of polynomial volume growth, associated with $\mu$ a left invariant sub-Laplacian and compared the associated heat kernel with $\mu$. Theorem 1.16 in there gives an estimate like the one you want: $$|\mu^{*n}(x, y) − p_n^{H_{\mu}}(x, y)| ≤ cn^{−(D+1)/2},$$ where $p_n^{H_\mu}$ is the heat kernel in question.