White Noise Driven Korteweg-de Vries Equation

We consider a stochastic Korteweg-de Vries equation on the real line. The noise is additive. We use function spaces similar to those introduced by Bourgain to prove well posedness results for the Korteweg-de Vries equation in L²(ℝ). We are able to handle a noise which is locally white in space and time. More precisely, it is a space-time white noise multiplied by an L²-function of the space variable. Due to the lack of a priori estimates, we can only get a local existence result in time. However, we obtain the global existence of L²(ℝ) solutions when the covariance operator of the noise is Hilbert-Schmidt in L²(ℝ).