Invariance of the White Noise for KdV

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space (b) over cap (s)(p, infinity), sp <-1, contains the support of the white noise. Then, we prove local well-posedness in (s)(p, infinity) for p = 2+, s = -1/2+ such that sp <-1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko [21].