Very Strong Disorder for the Parabolic Anderson model in low dimensions

TL;DR

This paper demonstrates that in low dimensions, the Parabolic Anderson Model exhibits very strong disorder with negative free energy, extending previous results and adapting proofs from discrete polymer models.

Contribution

It adapts Lacoin's proof for discrete polymers to show very strong disorder in the continuous Parabolic Anderson Model in low dimensions.

Findings

In dimension 1 and 2, the free energy is always negative.

Very strong disorder always holds in these low dimensions.

The proof adapts techniques from discrete polymer models.

Abstract

We study the free energy of the Parabolic Anderson Model, a time-continuous model of directed polymers in random environment. We prove that in dimension 1 and 2, the free energy is always negative, meaning that very strong disorder always holds. The result for discrete polymers in dimension two, as well as better bounds on the free energy on dimension 1, were first obtained by Hubert Lacoin, and the goal of this paper is to adapt his proof to the Anderson Parabolic Model.