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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