The Parabolic Anderson Model with Acceleration and Deceleration

TL;DR

This paper investigates the large-time behavior of the parabolic Anderson model with variable diffusion speed, revealing critical scales, phase transitions, and new variational problems that extend previous constant-speed results.

Contribution

It introduces a detailed analysis of acceleration and deceleration effects in the model, identifying critical scales and phases, and develops new variational methods for asymptotic analysis.

Findings

Identified a lower critical scale where mass flow gets stuck.

Discovered an upper critical scale where potential effects diminish.

Mapped out five distinct phases of the model's behavior.

Abstract

We describe the large-time moment asymptotics for the parabolic Anderson model where the speed of the diffusion is coupled with time, inducing an acceleration or deceleration. We find a lower critical scale, below which the mass flow gets stuck. On this scale, a new interesting variational problem arises in the description of the asymptotics. Furthermore, we find an upper critical scale above which the potential enters the asymptotics only via some average, but not via its extreme values. We make out altogether five phases, three of which can be described by results that are qualitatively similar to those from the constant-speed parabolic Anderson model in earlier work by various authors. Our proofs consist of adaptations and refinements of their methods, as well as a variational convergence method borrowed from finite elements theory.