  We construct dynamics for the defocusing real-valued (Miura) mKdV equation on
the real line with initial data distributed according to Gibbs measure. We also
prove that Gibbs measure is invariant under these dynamics. On the way, we
provide a new proof of the invariance of the Gibbs measure under mKdV on the
torus.
  Building on these results, we construct new measure-preserving dynamics for
the KdV equation on the whole real line. Samples from this family of measures
exhibit the same local regularity as white-noise, but completely different
statistics!
