Periodic Stochastic Korteweg-de Vries Equation

TL;DR

This paper proves local well-posedness for the periodic stochastic Korteweg-de Vries equation with additive white noise, using Besov-type spaces and Bourgain space variants to handle low regularity.

Contribution

It introduces a novel approach with Besov-type spaces and Bourgain space variants to establish well-posedness for stochastic KdV with white noise.

Findings

Proves local well-posedness in low regularity spaces.

Develops a nonlinear estimate on the second iteration.

Extends deterministic KdV results to stochastic setting.

Abstract

We prove the local well-posedness of the periodic stochastic Korteweg-de Vries equation with the additive space-time white noise. In order to treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space \hat{b}^s_{p, \infty}(T) for s= -1/2+, p = 2+ such that sp < -1. In establishing the local well-posedness, we use a variant of the Bourgain space adapted to \hat{b}^s_{p, \infty}(T) and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in M(T), the space of finite Borel measures on T.