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We study the stochastic Korteweg-de Vries equation (SKdV) with an additivespace-time white noise forcing, posed on the one-dimensional torus. Inparticular, we construct global-in-time solutions to SKdV with spatial whitenoise initial data. Due to the lack of an invariant measure, Bourgain'sinvariant measure argument is not applicable to this problem. In order toovercome this difficulty, we implement a variant of Bourgain's argument in thecontext of an evolution system of measures and construct global-in-timedynamics. Moreover, we show that the white noise measure with variance $1+t$ isan evolution system of measures for SKdV with the white noise initial data.

Global dynamics for the stochastic KdV equation with white noise as initial data