Probabilistic well-posedness for the cubic wave equation

TL;DR

This paper introduces a probabilistic notion of well-posedness for the cubic wave equation with random initial data, allowing more general randomizations and demonstrating favorable dynamical properties of the flow.

Contribution

It extends previous work by permitting broader randomizations and establishes a new probabilistic continuity for the flow of the cubic wave equation.

Findings

Allows general infinite product measures for randomization

Constructs a flow with probabilistic continuity

Demonstrates well-posedness in a probabilistic framework

Abstract

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs measures), and second, we show that the flow that we are able to construct enjoys very nice dynamical properties, including a new notion of probabilistic continuity.