Invariant measures for mKdV and KdV in infinite volume

We construct dynamics for the defocusing real-valued (Miura) mKdV equation onthe real line with initial data distributed according to Gibbs measure. We alsoprove that Gibbs measure is invariant under these dynamics. On the way, weprovide a new proof of the invariance of the Gibbs measure under mKdV on thetorus. Building on these results, we construct new measure-preserving dynamics forthe KdV equation on the whole real line. Samples from this family of measuresexhibit the same local regularity as white-noise, but completely differentstatistics!