Structure preservation and emergent dissipation in stochastic wave equations with transport noise

TL;DR

This paper investigates how transport noise affects nonlinear wave equations, showing that noise acting on displacement preserves structure while noise on velocity introduces dissipation, leading to different limiting behaviors.

Contribution

It provides a rigorous analysis of the effects of transport noise on nonlinear wave equations, including well-posedness and scaling limits, revealing structure preservation or dissipation.

Findings

Noise on displacement preserves the original wave structure.

Noise on velocity induces an effective Laplacian damping term.

Rescaled dynamics with velocity noise lead to a stochastic Westervelt-type model.

Abstract

We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative term. We establish well-posedness in both cases and analyse the associated scaling limits. When the noise acts on the displacement, the system preserves its original structure and converges to the deterministic nonlinear wave equation, whereas if it acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model.