The Burgers equation with Poisson random forcing

TL;DR

This paper studies the Burgers equation driven by Poissonian noise, establishing existence, uniqueness, and ergodic properties of solutions, and analyzing the behavior of global minimizers under random forcing.

Contribution

It introduces a novel analysis of the Burgers equation with Poissonian forcing, proving existence, uniqueness, and ergodic properties without periodicity assumptions.

Findings

Existence and uniqueness of global solutions under weak concentration conditions.

Identification of the main ergodic component for the model.

Demonstration of diffusive behavior of global minimizers in periodic forcing.

Abstract

We consider the Burgers equation on the real line with forcing given by Poissonian noise with no periodicity assumption. Under a weak concentration condition on the driving random force, we prove existence and uniqueness of a global solution in a certain class. We describe its basin of attraction that can also be viewed as the main ergodic component for the model. We establish existence and uniqueness of global minimizers associated to the variational principle underlying the dynamics. We also prove the diffusive behavior of the global minimizers on the universal cover in the periodic forcing case.