Existence of invariant measures for the stochastic damped KdV equation

TL;DR

This paper proves the existence of invariant measures for the stochastic damped KdV equation on the real line, demonstrating long-term statistical stability of solutions under stochastic forcing.

Contribution

It establishes the Feller property and proves the existence of invariant measures using asymptotic compactness and the Aldous criterion, advancing understanding of stochastic KdV dynamics.

Findings

Existence of invariant measures for the stochastic damped KdV.

Feller property of the solution semigroup.

Use of Aldous criterion for tightness and asymptotic compactness.

Abstract

We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.