KdV Preserves White Noise

2006/11/06 by Jeremy Quastel, Quastel, Jeremy, Benedek Valkó +1 · 1 citation
Mathematics · Physics and Astronomy · #35Q53 #60H40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.math/0611152

openalex publication_date 2006/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that white noise is an invariant measure for the Korteweg-deVries equation on \mathbb T. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain.

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