Well-posedness of the focusing stochastic nonlinear Schr\"odinger equation: $L^2$-critical and supercritical cases

TL;DR

This paper analyzes the well-posedness and blow-up phenomena of the focusing stochastic nonlinear Schrödinger equation in critical and supercritical regimes, providing probabilistic bounds and criteria for finite-time blow-up.

Contribution

It establishes conditions for global existence and finite-time blow-up of solutions in the stochastic setting, extending deterministic results to probabilistic frameworks.

Findings

Quantitative bounds on solution behavior over time.

Criteria for finite-time blow-up with positive probability.

Analysis of solution behavior under additive and multiplicative noise.

Abstract

We study the focusing $L^2$-critical and supercritical stochastic nonlinear Schr\"odinger equation subject to additive or multiplicative noise. We investigate global or long time behavior of solutions in $H^1$, which would correspond to global well-posedness in the deterministic case, with either deterministic or random initial data, and establish quantitative information about the well-posedness time, its probability and bounds on the solution in both cases. We then give criteria for finite time blow-up with positive probability for an $H^1$-valued initial data with positive energy in both cases.