Wiener Sausages in Riemann Surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:51:27Z http://mathoverflow.net/feeds/question/59177 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59177/wiener-sausages-in-riemann-surfaces Wiener Sausages in Riemann Surfaces ght 2011-03-22T12:12:08Z 2011-04-04T19:45:38Z <p>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 $r>0$. </p> <p>Define the Wiener sausage as:</p> <p>$$W_{r}(t):=\{ x\in M: d(x,W(s))\leq r\quad\text{for}\quad 0\leq s\leq t \}.$$</p> <p>It is known that in $\mathbb{R}^{2}$ and for t sufficiently large and $r$ fixed</p> <p>$$\mathbb{E}[\mathrm{vol}(W_{r}(t))]=\frac{2\pi t}{\log(t)}(1+o(1)).$$</p> <p>Is there any analogue result for a general Riemann surface or at least the hyperbolic space?</p> <p>Thanks!</p> <p>--Gabriel</p> http://mathoverflow.net/questions/59177/wiener-sausages-in-riemann-surfaces/60600#60600 Answer by ght for Wiener Sausages in Riemann Surfaces ght 2011-04-04T19:45:38Z 2011-04-04T19:45:38Z <p>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 <em>"The Wiener Sausages and a Theorem of Spitzer in Riemannian Manifolds"</em>, Probability and Harmonic Analysis, New York, pp. 45-60, 1986.</p>