Periodic Solutions of the Korteweg--de Vries Equation Driven by White Noise

Source abstract

We consider a Korteweg--de Vries equation perturbed by a noise term on a bounded interval with periodic boundary conditions. The noise is additive, white in time, and "almost white in space." We get a local existence and uniqueness result for the solutions of this equation. In order to obtain the result, we use the precise regularity of the Brownian motion in Besov spaces, and the method which was introduced by Bourgain, but based here on Besov spaces.