## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 1 (1994), 160-176.

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

Erwin Bolthausen and Frank Den Hollander

#### 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, $\gamma < 2/d, d \geq 2$.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 1 (1994), 160-176.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988853

**Digital Object Identifier**

doi:10.1214/aop/1176988853

**Mathematical Reviews number (MathSciNet)**

MR1258871

**Zentralblatt MATH identifier**

0793.60086

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J55: Local time and additive functionals

Secondary: 60G17: Sample path properties 60G57: Random measures

**Keywords**

Superprocesses measure-valued processes local times join continuity Hoder continuity path properties Haudorff dimension

#### Citation

Bolthausen, Erwin; Hollander, Frank Den. Survival Asymptotics for Brownian Motion in a Poisson Field of Decaying Traps. Ann. Probab. 22 (1994), no. 1, 160--176. doi:10.1214/aop/1176988853. https://projecteuclid.org/euclid.aop/1176988853