Let $M$ be a Riemann surface (or a higher dimensional manifold) and let's assume that it's geodesically complete. Let $W(t)$ be a Brownian motion on the surface accordingly to the manifold's Laplacian and let $r>0$.

Define the Wiener sausage as:

$$ W_{r}(t):=\{ x\in M: d(x,W(s))\leq r\quad\text{for}\quad 0\leq s\leq t \}. $$

It is known that in $\mathbb{R}^{2}$ and for t sufficiently large and $r$ fixed

$$ \mathbb{E}[\mathrm{vol}(W_{r}(t))]=\frac{2\pi t}{\log(t)}(1+o(1)). $$

Is there any analogue result for a general Riemann surface or at least the hyperbolic space?

Thanks!

--Gabriel