Well-posedness of the focusing stochastic nonlinear Schrödinger equation: $L^2$-critical and supercritical cases

TL;DR

This work analyzes the focusing stochastic nonlinear Schrödinger equation with either additive or multiplicative noise in the $L^2$-critical and intercritical regimes. It develops local well-posedness theory in $H^1$ for both noise types, derives mass and energy evolution laws under stochastic perturbations, and leverages ground-state thresholds to distinguish global existence from blow-up in a probabilistic sense. For the $L^2$-critical case, sub-threshold mass yields almost-sure global existence (multiplicative noise) or long-time control (additive noise); for intercritical data, the paper proves a probabilistic global-in-time result up to a finite time $T^*$ and, under small noise, positive-probability blow-up, with explicit dependence on noise strength. The results extend deterministic global-well-posedness and blow-up dichotomies to stochastic settings and quantify how noise modulates the time scales and likelihoods of global behavior vs finite-time singularity.

Abstract

We study the focusing $L^2$-critical and supercritical stochastic nonlinear Schrödinger 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.

Figures (1)

• Figure 1: Comparison of deterministic (left) and stochastic (right) cases in the intercritical setting: $g(x) = \frac{1}{2} x^2$, $f(x) = \frac{1}{2} (x^2-Bx^{n\sigma})$, $x^{\ast} = \|\nabla Q\|_{L^2} \|Q\|_{L^2}^\alpha$, $f(x^\ast) = H(Q)M(Q)^\alpha$. In the deterministic case initially starting at time $t=0$ at $x = \|\nabla u(t)\|_{L^2} \|u_0\|_{L^2}^\alpha$, the solution stays on the green line $y_0 = H(u_0)M(u_0)^\alpha$ (conserved in time), meaning that $\|\nabla u(t)\|_{L^2}$ is bounded for all time $t$, being trapped on the left of $f$ (and thus, globally well-posed), while in the stochastic case starting at $X(0) = \|\nabla u_0\|_{L^2} \|u_0\|^\alpha_{L^2}$, $Y(0) = H(u_0) M(u_0)^\alpha$, the solution stays in the purple area up to time $\tau_\delta: = \inf\{s \geq 0: H(u(s))M(u_0)^\alpha \geq \delta f(x^\ast)\}$ with $X(\tau_\delta) = \sup_{s \leq \tau_\delta} \|\nabla u(s)\|_{L^2} \|u_0\|^\alpha_{L^2}$, $Y(\tau_\delta) = \sup_{s \leq \tau_\delta} H(u(s)) M(u_0)^\alpha \leq \delta f(x^\ast)$.

Theorems & Definitions (44)

• Theorem 1.1: HR2007HR2008DHR
• Theorem 3.1: Kwa_Szy
• Remark 3.2
• Theorem 3.3
• Lemma 3.4
• Remark 3.5
• proof : Proof of Lemma \ref{['time_evol_H_2.4']}
• Theorem 3.6
• proof
• Theorem 3.7