Random data theories for PDE’s were first developed by Bourgain in the 90’ in the context of his seminal works on Gibb’s measures for non linear Schroedinger equations. In Bourgain’s approach, randomness was an enemy because it forced to work at very rough regularity levels. It was only 15 years later that we realised with Tzvetkov that far from being an enemy, randomness could actually help in a PDE context and allow to exhibit examples where, while deterministically solutions to PDE’s could exhibit bad behaviours, these pathological behaviours were (in some cases) actually quite rare as for suitable natural probability measures on the set of initial, they, almost surely, do not happen. I will present in this talk some basic ideas on this theory and some striking recent results (with their deterministic counterparts).
