Weak pullback attractors for damped stochastic fractional Schr\"odinger equation on $\mathbb{R}^n

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Abstract

This article discusses the weak pullback attractors for a damped stochastic fractional Schr\"odinger equation on $\mathbb{R}^n$ with $n\geq 2$. By utilizing the stochastic Strichartz estimates and a stopping time technique argument, the existence and uniqueness of a global solution for the systems with the nonlinear term $|u|^{2\sigma}u$ are proven. Furthermore, we define a mean random dynamical system due to the uniqueness of the solution, which has a unique weak pullback mean random attractor in $L^\rho\left(\Omega; L^2\left(\mathbb{R}^n\right)\right)$. This result highlights the long-term dynamics of a broad class of stochastic fractional dispersion equations.