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

2024/06/27 by Brzeźniak, Zdzisław, Ferrario, Benedetta, Maurelli, Mario +1
#35Q55 #35R60 #60G10 #60H15 #60H30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2406.19214

Abstract

We consider the nonlinear Schrödinger equation on the d-dimensional torus \mathbb T^d, with the nonlinearity of polynomial type |u|^2σu. For any σ∈ \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.

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