Mean Attractors and Invariant Measures for Fractional Stochastic Lattice Systems Driven by Nonlinear Noise

• Published in: Journal of theoretical probability Vol. 38; no. 4
• Main Authors: Chen, Pengyu, Bai, Ailin, Zhang, Xuping
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2025; Springer Nature B.V
• Subjects: Invariants; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Statistics; Tightness; Uniqueness; 37L55; Invariant measure; 37B55; Nonlinear noise; Fractional stochastic lattice system; Mean random attractors; Tightness; 35B40; 60H15
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-025-01442-9

This paper is concerned with the existence of mean random attractors and invariant measures for fractional stochastic lattice systems driven by nonlinear noise. We firstly establish the global existence and uniqueness of solutions, and then prove the existence and uniqueness of weak pullback mean random attractors of the fractional stochastic lattice systems in the Bochner space L 2 ( Ω , ℓ 2 ) . Under certain conditions, we establish the tightness of a family of distributions of solutions by using the uniform estimates on the tails of solutions and then show the existence of invariant measures of the system. We also discuss the limiting behavior of invariant measures of fractional stochastic lattice systems driven by nonlinear noise as ϵ → 0 . Finally, under further assumptions on the nonlinear terms, we show that the system has a unique, ergodic, mixing, and stable invariant probability measure in ℓ 2 .