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Asymptotics for the Wiener sausage with drift. (English) Zbl 0586.60076
A particle is considered which moves in \({\mathbb{R}}^ d\) according to a Brownian motion with drift \(h\neq 0\). The space is assumed to contain random traps. The probability of survival of the particle up to time T decays exponentially as \(T\to \infty\) with a positive decay rate \(\lambda\). \(\lambda\) is shown to be a non-analytic function of \(| h|\). For small \(| h|\) the decay rate is given by \(\lambda (h)=2^{-1}| h|^ 2\); but if \(| h|\) exceeds a certain critical value, \(\lambda\) (h) depends also on the parameters describing trapping. Upper and lower bounds for \(\lambda\) (h) are given, which imply the asymptotic linearity of \(\lambda\) (h) for large \(| h|.\)
The critical point marks a transition from localized to delocalized behavior. A variational formula for the decay rate is given on the level of generalized processes, which elucidates the mathematical mechanism behind observations made earlier by Grassberger and Procaccia on the basis of computer simulations.

60J65 Brownian motion
60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI
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