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