A factorization formula for the partition function in the semi-discrete parabolic Anderson model

TL;DR

This paper proves the existence and positivity of limits of the partition function for a random walk in a Wiener potential in high-temperature regime, and introduces a factorization formula valid up to sub-ballistic scales.

Contribution

It establishes a new factorization formula for the point-to-point partition function in the semi-discrete parabolic Anderson model.

Findings

Limits of the partition function exist in $L^2$ and almost surely.

Limiting partition functions are positive almost surely.

Factorization formula holds up to sub-ballistic scale.

Abstract

We consider a continuous-time simple symmetric random walk on the integer lattice $\mathbb{Z}^d$ in dimension $d \geq 3$, subject to a random potential given by a field of two-sided Wiener processes. In the high-temperature regime, we prove the existence of the $L^2$- and almost sure limits of the partition function as time $t \to \pm \infty$, and show that these limiting partition functions are positive almost surely. Our main result is a factorization formula for the point-to-point partition function, which is shown to be valid up to any sub-ballistic scale.