Weak mean attractors and invariant measures for stochastic Schrödinger delay lattice systems

• Published in: Journal of dynamics and differential equations Vol. 35; no. 4; pp. 3201 - 3240
• Main Authors: Chen, Zhang, Wang, Bixiang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Applications of Mathematics; Hilbert space; Invariants; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Tightness; Time lag; 37L55; 60H10; Invariant measure; Asymptotic compactness; Stochastic Schrödinger lattice system; Weak pullback mean attractor; Time delay; Tail-estimate; 37L40
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-021-10085-3

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system.