White noise for KdV and mKdV on the circle

Details White noise for KdV and mKdV on the circle White noise for KdV and mKdV on the circle

2010-07-12

arxiv.org/abs/1007.1954v1

Mathematics

Analysis of PDEs

26 pages. To appear in RIMS Kokyuroku Bessatsu

Scientific paper

We survey different approaches to study the invariance of the white noise for the periodic KdV. We mainly discuss the following two methods. First, we discuss the PDE method, following Bourgain \cite{BO4}, in a general framework. Then, we show how it can be applied to the low regularity setting of the white noise for KdV by introducing the Besov-type space \hat{b}^s_{p, \infty}, sp< -1. Secondly, we describe the probabilistic method by Quastel, Valk\'o, and the author \cite{OQV}. We also use this probabilistic approach to study the white noise for mKdV.

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