Survival Asymptotics for Brownian Motion in a Poisson Field of Decaying Traps

Abstract

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 $t \rightarrow \nu (t)$ nonincreasing and $nu (t) \sim \nu t^{-\gamma}, t \rightarrow \infty$, have a large deviation behavior similar to that of $|W(t)|$ established by Donsker and Varadhan. 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 crossover at $\gamma = 2/d$; (ii) in $d \geq 1$ the survival strategy changes at time $s = \lbrack\gamma/(1 + \gamma)\rbrack t$, provided $\gamma < 1/2, d = 1$, respectively, $\gamma < 2/d, d \geq 2$.