Brownian motion in a Poissonian potential
We study here a \(d\)-dimensional Brownian motion in a random potential \(V (\cdot, \omega)\) obtained as the sum of translations of a given fixed nonnegative shape function at the points of a Poisson cloud of constant intensity \(v\). We are interested in the large \(t\) behavior for typical cloud configurations, of the Brownian path in time \(t\) under the influence of the natural Feynman-Kac weight associated to \(V(\cdot,\omega)\). In particular, we show that the location at time \(t\) of the process tends to be concentrated near points of suitably ``low local eigenvalue'' of \(-{1 \over 2} \Delta+V (\cdot, \omega)\), which lie almost at distance \(t\) from the origin. Near these points one can find in the cloud a ``big hole'' or ``claring'' of size \(\sim \text{const} (\log t)^{1/d}\) with volume like a ball of radius \(R_ 0(d,v) (\log t)^{1/d}\).
scientific article; zbMATH DE number 568998