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KdV Preserves White Noise

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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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Authors and Affiliations

  1. Departments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada, M5S2E4

    Jeremy Quastel & Benedek Valkó

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  1. Jeremy Quastel
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  2. Benedek Valkó
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Correspondence to Jeremy Quastel.

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Communicated by A. Kupiainen.

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Quastel, J., Valkó, B. KdV Preserves White Noise. Commun. Math. Phys. 277, 707–714 (2008). https://doi.org/10.1007/s00220-007-0372-6

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  • DOI: https://doi.org/10.1007/s00220-007-0372-6

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