Unconditional well-posedness of the stochastic Korteweg-de Vries equation on the real line

Abstract

We study well-posedness issues of the stochastic Korteweg-de Vries equation (SKdV) with an additive noise, posed on the real line. By using the Fourier restriction norm method adapted to the Fourier-Lebesgue space in time, we first prove global well-posedness of SKdV in $L^2(\mathbb R)$ without assuming the homogenous Sobolev regularity, which was imposed in a work by de Bouard, Debussche, and Tsutsumi (1999). Then, by adapting the argument by Zhou (1997) to the stochastic setting, we prove optimal pathwise unconditional uniqueness for SKdV in $L^2(\mathbb R)$. In the appendix, we present a short argument for proving boundedness of the multiplication by a sharp cutoff function in the Fourier-Lebesgue and Sobolev spaces, which is of interest in its own right.