Global dynamics for the stochastic KdV equation with white noise as initial data

TL;DR

This paper establishes the existence of global solutions for the stochastic KdV equation with white noise initial data on a torus, using a novel approach involving evolution systems of measures.

Contribution

It introduces a new method to construct global solutions for SKdV with white noise initial data without relying on invariant measures.

Findings

White noise measure with variance 1+t is an evolution system of measures for SKdV.

Constructed global-in-time solutions for SKdV with white noise initial data.

Developed a variant of Bourgain's argument suitable for evolution systems of measures.

Abstract

We study the stochastic Korteweg-de Vries equation (SKdV) with an additive space-time white noise forcing, posed on the one-dimensional torus. In particular, we construct global-in-time solutions to SKdV with spatial white noise initial data. Due to the lack of an invariant measure, Bourgain's invariant measure argument is not applicable to this problem. In order to overcome this difficulty, we implement a variant of Bourgain's argument in the context of an evolution system of measures and construct global-in-time dynamics. Moreover, we show that the white noise measure with variance $1+t$ is an evolution system of measures for SKdV with the white noise initial data.