2023/03/07 by Wenjie Hu, Hu, Wenjie, Tomás Caraballo +1
Economics, Econometrics and Finance · Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2303.04102
openalex publication_date 2023/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main purpose of this work is to characterize the almost sure local structure stability of solutions to a class of linear stochastic partial functional differential equations (SPFDEs) by investigating the Lyapunov exponents and invariant manifolds near the stationary point. It is firstly proved that the trajectory field of the stochastic delayed stochastic partial functional differential equation admits an almost sure continuous version which is compact for t>τ by a delicate construction based on the random semiflow generated by the diffusion term. Then it is proved that the version generates a random dynamical system(RDS) by the Wong-Zakai approximation of the stochastic partial differential equation constructed by the diffusion term. Subsequently, it is shown that the constructed linear cocycle admits fixed (at most) countable set of Lyapunov exponents and the associate Oseledets random filtration of the Banach space is obtained by adopting the infinite-dimensional multiplicative ergodic theorem in Banach spaces established by Lian and Lu [Mem Amer Math Soc, 2010, 206: 967]. As a by product, the stable-manifolds theorem for the linear SPFDE in the hyperbolic case is also established.