Math Colloquium: Probability in the PDE theory

In this talk, we discuss how probabilistic ideas are applied to study PDEs. First, we briefly go over the basic theory of Gaussian Hilbert spaces and abstract Wiener spaces to determine function spaces which capture the regularity of the Brownian motion and the white noise. Next, we go over Bourgain's idea to establish the invariance of Gibbs measures for PDEs. We then establish local well-posedness (LWP) of KdV with the white noise as initial data via the second iteration introduced by Bourgain. This in turn provides almost sure global well-posedness (GWP) of KdV as well as the invariance of the white noise. Then, we discuss how one can use the same idea to obtain LWP of the stochastic KdV with additive space-time (non-smoothed) white noise in the periodic setting.

We also consider the weak convergence problem of the grand canonical ensemble (i.e. the interpolation measure of the usual Gibbs measure and the white noise) with a small parameter (tending to 0) to the white noise. This result, combined with the GWP in $H^{-1}$ by Kappeler and Topalov, provides another proof of the invariance of the white noise for KdV. In this talk, we discuss the same weak convergence problem for mKdV and cubic NLS, which provides the ``formal'' invariance of the white noise. This part is a joint work with J. Quastel and B. Valk\'o.

Lastly, if time permits, we discuss the well-posedness of the Wick ordered cubic NLS on the Gaussian ensembles below $L^2$. The main ingredient is nonlinear smoothing under randomization of initial data. For GWP, we also use the invariance (of the Gaussian ensemble) under the linear flow. This part is a joint work with J. Colliander.