Invariant Measure and Random Attractors for Stochastic Differential Equations with Delay

• Shangjiang Guo ^[1] ; Shangzhi Li ^[1]
  1. [1] China University of Geosciences

     China University of Geosciences

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 2, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we consider the long term behavior of solutions to stochastic delay differential equations with additive noise. We first establish the existence of an invariant measure by regarding the stochastic equation as a perturbation of a deterministic equation. If the deterministic part is stable it is plausible to expect the existence of an invariant measure under some conditions on the perturbation. Different from traditional methods, a retarded Ornstein-Uhlenbeck process is introduced to the study of the existence of a continuous random dynamical system and tempered pullback attractors. In particular, we prove pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors. We also establish the upper semicontinuity of the random attractors when noise intensity approaches zero, and investigate the exponential stability of stationary solutions in the mean square sense and almost surely. It is found that our theoretical results and methods can be generalized to other stochastic delay differential equations, even with multiplicative noise. Finally, our theoretical results are illustrated by an applications to a stochastic Nicholson’s blowflies model.

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