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Brownian survival among Gibbsian traps. (English) Zbl 0769.60104
The author studies the long time survival probability of a Brownian motion \(Z\) on \(R^ d\) \((d\geq 1)\) which moves among random obstacles constructed by translating a model nonpolar compact set at the points of a Gibbs point process independent of \(Z\). It is assumed that the law \(\tilde P\) of this point process satisfies the Dobrushin-Lanford-Ruelle equations relative to an activity number \(\nu>0\) and a suitable transformation invariant pair potential \(V(x-y)\). Let \(T\) denote the entrance time of \(Z\) into the obstacles and let \(P_ 0\) denote standard Wiener measure. The author shows that under additional assumptions on \(\tilde P\), the limit \[ \lim_{t\to\infty}t^{-d/(d+2)}\log(\tilde P\otimes P_ 0(T>t))=-c(d,p) \] exists where \(c(d,p)\) depends on a certain pressure \(p\in(0,\infty)\) whose existence follows from the hypotheses on \(V(\cdot)\).

60K40 Other physical applications of random processes
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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