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)]^s_p,￥{widehat{b}^s_{p,infty}} , sp < −1, contains the support of the white noise. Then, we prove local well-posedness in [^(b)]^s_{p, ￥}{widehat{b}^s_{p, infty}} for p = 2 + , s = -frac12+{s = -frac{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].