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



Wiener Sausages in Riemann Surfaces
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.


