Invariance of the white noise for KdV

TL;DR

This paper proves the invariance of mean zero white noise under the periodic KdV flow by establishing local well-posedness in a specific Besov-type space and using a weighted Bourgain space variant.

Contribution

It provides an analytical proof of white noise invariance for KdV, extending previous results by identifying the support in a Besov space and proving well-posedness there.

Findings

White noise support in ^s_{p,} space for sp<-1

Local well-posedness in ^s_{p,} with p=2+, s=-1/2+

Analytical proof of white noise invariance under KdV flow

Abstract

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space \hat{b}^s_{p, \infty}, sp <-1, contains the support of the white noise. Then, we prove local well-posedness in \hat{b}^s_{p, \infty} 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.