Invariant Gibbs measure for Anderson nonlinear wave equation

TL;DR

This paper constructs an invariant Gibbs measure for a nonlinear wave equation influenced by Anderson Hamiltonian, employing stochastic control and renormalization techniques to analyze the measure's properties and invariance.

Contribution

It introduces a novel coupling of the Anderson Hamiltonian-based Gaussian measure with the Gaussian free field, enabling the definition of renormalized powers and proving invariance of the Gibbs measure.

Findings

Established the existence of a regular coupling between the Anderson measure and Gaussian free field.

Defined renormalized powers of the Anderson free field.

Proved invariance of the quartic Gibbs measure under the nonlinear wave flow.

Abstract

We study the Gaussian measure whose covariance is related to the Anderson Hamiltonian operator, proving that it admits a regular coupling to the (standard) Gaussian free field exploiting the stochastic optimal control formulation of Gibbs measures. Using this coupling, we define the renormalized powers of the Anderson free field and we prove that the associated quartic Gibbs measure is invariant under the flow of a nonlinear wave equation with renormalized cubic nonlinearity.