Random weighted Sobolev inequalities

We extend a randomisation method, introduced by Burq-Lebeau on compact manifolds, to the case of the harmonic oscillator. We construct measures, under Log-Sobolev type assumptions, on the support of which we prove optimal weighted Sobolev estimates on $\mathbb R^d$. As an application we can prove almost sure global well posedness results for the nonlinear Schrödinger equation with harmonic potential.

This is a joint work with Aurélien Poiret and Didier Robert.