Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity

Abstract

We investigate the invariance of the Gibbs measure for the fractional Schrödinger equation of exponential type (expNLS) i∂tu+(−Δ)α2u=2γβeβ|u|2u on d-dimensional compact Riemannian manifolds Md, for a dispersion parameter α>d, some coupling constant β>0, and γ≠0. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case γ>0, the measure is well-defined in the whole regime α>d and β>0 (Theorem 1.1 (i)), while in the focusing case γ<0 its partition function is always infinite for any α>d and β>0, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure, in the defocusing case γ>0, in the whole regime α>d and β>0 (Theorem 1.4 (i)). In the large dispersion regime α>2d, we can improve this result by constructing a local deterministic flow for (expNLS) for any β>0 and γ≠0. Using the Gibbs measure, we prove that solutions are almost surely global for any β,γ>0, and that the Gibbs measure is invariant (Theorem 1.4 (ii)). (iii) Finally, in the particular case d=1 and M=T, we are able to exploit some probabilistic multilinear smoothing effects to build a global probabilistic flow for (expNLS) for 1+22<α≤2 for any β,γ>0 (Theorem 1.6).