In this talk we show how certain well-posedness results that are not available using only deterministic techniques (eg. Fourier and harmonic analysis) can be obtained when introducing randomization in the set of initial data and using powerful but still classical tools from probability as well. These ideas go back to seminal work by J. Bourgain on the invariance of Gibbs measures associated to dispersive PDE. We will first explain some of these ideas and review some recent probabilistic well-posedness results for NLS. We will then describe recent work of myself joint with Chanillo, Czubak, Mendelson and Staffilani in which we treat probabilistic well-posedness of a geometric wave equation with randomized supercritical data. If time permits, we will discuss a work in progress about non-equilibrium invariant measures for resonant NLS (joint with Hani, Mattingly, Rey-Bellet and Staffilani).
Speaker(s): Andrea Nahmod (UMASS, Amherst)