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

TL;DR

This paper investigates the existence of global attractors for the damped stochastic fractional Schrödinger equation on Euclidean space, establishing well-posedness and the dynamical system's long-term behavior under Gaussian noise.

Contribution

It proves the existence and uniqueness of solutions and demonstrates the presence of a global attractor for the stochastic fractional Schrödinger equation with damping.

Findings

Existence and uniqueness of global solutions in $H^{eta}( ext{R}^n)$.

The stochastic fractional Schrödinger equation generates a dynamical system.

Existence of a global attractor in the solution space.

Abstract

We study the random attractors associated with the stochastic fractional Schr\"odinger equation on $\mathbb{R}^n$. Utilizing the stochastic Strichartz estimates for the damped fractional Schr\"odinger equation with Gaussian noise, we show the existence and uniqueness of a global solution to the damped stochastic fractional nonlinear Schr\"odinger equation in $H^{\alpha}(\mathbb{R}^n)$. Furthermore, we demonstrate that this equation defines an infinite-dimensional dynamical system, which possesses a global attractor in $H^{\alpha}(\mathbb{R}^n)$.