Sausage Volume of the Random String and Survival in a medium of Poisson Traps

TL;DR

This paper derives asymptotic bounds on the survival probability of a stochastic heat equation-based polymer in Poisson trap environments, revealing decay rates similar to Brownian motion and analyzing the Wiener sausage growth around the polymer.

Contribution

It provides the first asymptotic bounds for the survival probability of a stochastic heat equation polymer in Poisson traps, including bounds for Wiener sausage growth.

Findings

Survival probability decays exponentially with rate proportional to T^{d/(d+2)}.

Bounds depend on polymer length J with different powers in upper and lower bounds.

Results extend understanding of polymer survival in random environments modeled by stochastic PDEs.

Abstract

We provide asymptotic bounds on the survival probability of a moving polymer in an environment of Poisson traps. Our model for the polymer is the vector-valued solution of a stochastic heat equation driven by additive spacetime white noise; solutions take values in ${\mathbb R}^d, d \geq 1$. We give upper and lower bounds for the survival probability in the cases of hard and soft obstacles. Our bounds decay exponentially with rate proportional to $T^{d/(d+2)}$, the same exponent that occurs in the case of Brownian motion. The exponents also depend on the length $J$ of the polymer, but here our upper and lower bounds involve different powers of $J$. Secondly, our main theorems imply upper and lower bounds for the growth of the Wiener sausage around our string. The Wiener sausage is the union of balls of a given radius centered at points of our random string, with time less than or equal to a given value.