Let $(M,g)$ a smooth closed Riemannian manifold with non trivial first homology group $H^1(M,\mathbb{R})$. Any element of $H^1(M,\mathbb{Z})$ will define a riemannian $\mathbb{Z}$-cover of $M$ denoted by $M_{\mathbb{Z}}$. Let us fix a base point $x_0$ on $M_{\mathbb{Z}}$ and consider the Brownian motion starting at $x_0$ and a ball $B$ of radius $C >0$ of center $x_0$.

Question. Does the probability of such a process to come back in the ball $B$ in time $t$ is like $\sqrt{t}$ ?

As it is true for $\mathbb{R}$ endowed with the flat metric I wish the same result holds also in this set up. A $\mathbb{Z}$-cover should look like $\mathbb{R}$ in large scale.