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 for , such that . In establishing local well-posedness, we use a variant of the Bourgain space adapted to 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 , the space of finite Borel measures on .
"Periodic stochastic Korteweg–de Vries equation with additive space-time white noise." Anal. PDE 2 (3) 281 - 304, 2009. https://doi.org/10.2140/apde.2009.2.281