Survival asymptotics for Brownian motion in a Poisson field of decaying traps. (English) Zbl 0793.60086

Bolthausen, Erwin; den Hollander, Frank

Ann. Probab. 22, No. 1, 160-176 (1994).

Summary: Let \(W(t)\) be the Wiener sausage in \(\mathbb{R}^ d\), that is, the \(a\)- neighborhood for some \(a>0\) of the path of Brownian motion up to time \(t\). It is shown that integrals of the type \(\int^ t_ 0\nu(s)d| W(s)|\), with \(t\to\nu(t)\) nonincreasing and \(\nu(t)\sim\nu t^{- \gamma}\), \(t\to\infty\), have a large deviation behavior similar to that of \(| W(t)|\) established by M. D. Donsker and S. R. S. Varadhan [Commun. Pure Appl. Math. 28, 525-565 (1975; Zbl 0333.60077)]. Such a result gives information about the survival asymptotics for Brownian motion in a Poisson field of spherical traps of radius \(a\) when the traps decay independently with lifetime distribution \(\nu(t)/\nu(0)\). There are two critical phenomena: (i) in \(d\geq 3\) the exponent of the tail of the survival probability has a cross-over at \(\gamma=2/d\); (ii) in \(d\geq 1\) the survival strategy changes at time \(s=[\gamma/(1+\gamma)]t\), provided \(\gamma<1/2\), \(d=1\), respectively, \(\gamma<2/d\), \(d\geq 2\).

MSC:

60J55	Local time and additive functionals
60G17	Sample path properties
60G57	Random measures

Keywords:

local times; Hölder continuity; path properties; Hausdorff dimension; Wiener sausage; asymptotics for Brownian motion in a Poisson field; traps; critical phenomena; survival strategy

Citations:

Zbl 0333.60077