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

Siva Athreya, Mathew Joseph, Carl Mueller

Abstract

In [AJM26], we gave large-time asymptotic bounds on the annealed survival probability of a moving polymer taking values in ${\mathbb R}^d, d \geq 1$. This polymer is a solution of a stochastic heat equation driven by additive spacetime white noise on $[0,T] \times [0,J]$, in an environment of Poisson traps. For fixed $J$, the annealed survivial probability decays exponentially with rate proportional to $T^{d/(d+2)}$. In this work we examine the large $J$ asymptotics of the annealed survival probability for any fixed time $T>0$. We prove upper and lower bounds for the annealed survival probability in the cases of hard obstacles. Our bounds decay exponentially with rate proportional to $J^{d/(d+2)}$. The exponents also depend on time $T >0$.

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

Abstract

In [AJM26], we gave large-time asymptotic bounds on the annealed survival probability of a moving polymer taking values in . This polymer is a solution of a stochastic heat equation driven by additive spacetime white noise on , in an environment of Poisson traps. For fixed , the annealed survivial probability decays exponentially with rate proportional to . In this work we examine the large asymptotics of the annealed survival probability for any fixed time . We prove upper and lower bounds for the annealed survival probability in the cases of hard obstacles. Our bounds decay exponentially with rate proportional to . The exponents also depend on time .
Paper Structure (7 sections, 12 theorems, 104 equations)

This paper contains 7 sections, 12 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Overview of Proof
  4. Preliminaries
  5. Brownian Bridge Estimates
  6. Bounds on the Volume of a Sausage
  7. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Consider the solution to eq:she with $d\ge 2$ and $T\geq 1$, and let $\nu$ and $a$ be as above. Assume $\textnormal{H}\equiv\infty$ on $B(\mathbf{0},a)$ and $\textnormal{H}\equiv0$ on $B(\mathbf{0},a)^c$. Then

Theorems & Definitions (24)

  • Theorem 1.1: Hard obstacles
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 14 more