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) $i\partial_t u + (-\Delta)^{\frac{\alpha}2} u = 2\gamma\beta e^{\beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $\mathcal{M}$, for a dispersion parameter $\alpha>d$, some coupling constant $\beta>0$, and $\gamma\neq 0$. (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 $0<\beta^\star_\alpha\sim (\alpha-d)$ (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 $0<\beta \ll\beta^\star_\alpha$, 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).