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On long excursions of Brownian motion among Poissonian obstacles. (English) Zbl 0760.60027
Stochastic analysis, Proc. Symp., Durham/UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 353-375 (1991).
Consider random obstacles in $${\mathbb{R}}^ d$$, given by means of closed balls of a fixed radius $$a>0$$, centered at the points of a Poisson cloud with constant intensity $$\nu>0$$. Let $$\{Z_ u\}_{u\geq 0}$$ be an independent Brownian motion starting from the origin and let $$T$$ be its entrance time into the obstacles. S. R. S. Donsker and M. D. Varadhan showed [Commun. Pure Appl. Math. 28, 525-565 (1975; Zbl 0333.60077)] that the probability of the event $$\{T>t\}$$ behaves like $$\exp[-t^{d/(d+2)}c(d,\nu)(1+o(1))]$$, independent of the radius $$a$$. This behaviour corresponds to the event that a ball of radius $$R_ 0 t^{1/(d+2)}$$ centered at the origin is free of obstacles and contains the path of the Brownian motion up to time $$t$$. The author considers long excursions up to time $$t$$: He proves upper and lower asymptotic bounds (depending on $$a$$ and $$x\geq 0$$) for the probability of the event $$\{ \sup_{u\in[0,t]}Z_ u>xt^{d/(d+2)}, T>t \}$$ as $$t\to\infty$$. The lower bound is derived by considering the event that the Brownian motion rushes in a time of order $$t^{d/(d+2)}$$ from the origin through a long cylindrical tube of approximate length $$xt^{d/(d+2)}$$ and then spends the remaining time until $$t$$ in a ball of radius $$R_ 0t^{1/(d+2)}$$ at the end of this tube. The upper bound is more difficult to prove; the author uses a technique which he developed in his earlier papers about the Wiener sausage.
For the entire collection see [Zbl 0768.60066].

##### MSC:
 60F10 Large deviations 60J65 Brownian motion