Abstract

In 1994, Bourgain constructed invariant Gibbs measures for NLS on the circle. Then, in 2000, he considered the limit of these invariant statistics, by taking larger and lager periods, and constructed unique solutions for the defocusing (sub-)cubic NLS on the real line. His result, however, focuses on the construction of solutions and does not discuss the limiting Gibbs measures on the real line or their invariance.

In this talk, we construct Gibbs measures for the defocusing NLS on the real line as a stationary diffusion process in x. Then, we show that these Gibbs measures are invariant for the defocusing (sub-)quintic NLS on the real line. We also discuss the limit Gibbs measures for the Dirichlet boundary value problem on the real line as well as the half line, allowing us to construct new rough solutions in these settings.

This is a joint work with Jeremy Quastel (University of Toronto) and Philippe Sosoe (Harvard University).