KdV Preserves White Noise
  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. 
Jeremy Quastel, Benedek Valko