Invariant Gibbs measure for a Schrodinger equation with exponential nonlinearity

We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $ipartial_t u + (-Delta)^{frac{alpha}2} u = 2gammabeta e^{beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $mathcal{M}$, for a dispersion parameter $alpha>d$, some coupling constant $beta>0$, and $gamma eq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $gamma>0$, the measure is well-defined in the whole regime $alpha>d$ and $beta>0$ (Theorem 1.1 (i)), while in the focusing case $gamma<0$ its partition function is always infinite for any $alpha>d$ and $beta>0$, 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 whole regime $alpha>d$ and $0<beta < beta^star_alpha$ for some natural parameter $0<beta^star_alphasim (alpha-d)$ (Theorem 1.3 (i)). In the large dispersion regime $alpha>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $beta>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<beta llbeta^star_alpha$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $mathcal{M}=mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+frac{sqrt{2}}2<alpha leq 2$, locally for arbitrary $beta>0$ and globally for $0<beta ll beta^star_alpha$ (Theorem 1.5).