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).