Periodic stochastic Korteweg–de Vries equation with additive space-time white noise

Abstract

We prove the local well-posedness of the periodic stochastic Korteweg–de Vries equation with the additive space-time white noise. To treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space ${\widehat{b}}_{p,\infty }^s\left(\mathbb{T}\right)$ for $s=-\frac{1}{2}+$, $p=2+$ such that $sp<-1$. In establishing local well-posedness, we use a variant of the Bourgain space adapted to ${\widehat{b}}_{p,\infty }^s\left(\mathbb{T}\right)$ 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 $M\left(\mathbb{T}\right)$, the space of finite Borel measures on $\mathbb{T}$.