TITLE: Wiener Sausages in Riemann Surfaces
QUESTION [11 upvotes]: 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

REPLY [3 votes]: I just found out that the case $r$ fixed and $t\to\infty$ for simply connected symmetric manifolds of non-positive sectional curvature and dimension $d\geq 3$, and strictly negative curvature for dimension $d=2$, was solved by Chavel and Feldman in "The Wiener Sausages and a Theorem of Spitzer in Riemannian Manifolds", Probability and Harmonic Analysis, New York, pp. 45-60, 1986.