White noise for KdV and mKdV on the circle
Abstract
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 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 bsp, ∞, sp< -1. Secondly, we describe the probabilistic method by Quastel, Valk\'o, and the author OQV. We also use this probabilistic approach to study the white noise for mKdV.
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