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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 $$V(\cdot)$$.

MSC:
 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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