Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schr\"odinger equation with multiplicative noise and arbitrary power of the nonlinearity

TL;DR

This paper proves global well-posedness and ergodic properties for the nonlinear Schrödinger equation with multiplicative noise on a torus, showing noise prevents finite-time blow-up and establishing invariant measures.

Contribution

It establishes the existence and uniqueness of global solutions and invariant measures for the stochastic nonlinear Schrödinger equation with arbitrary polynomial nonlinearity power.

Findings

Noise prevents finite-time blow-up.

Existence of invariant measures.

Uniqueness of invariant measures under certain conditions.

Abstract

We consider the nonlinear Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2\sigma}u$. For any $\sigma \in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in $H^s(\mathbb T^d)$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover we prove existence of invariant measures and their uniqueness under more restrictive assumptions on the noise term.