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A dynamical theory for singular stochastic delay differential equations i: linear equations and a multiplicative ergodic theorem on fields of banach spaces

Ghani Varzaneh, M ; Sharif University of Technology | 2022

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  1. Type of Document: Article
  2. DOI: 10.1137/21M1433435
  3. Publisher: Society for Industrial and Applied Mathematics Publications , 2022
  4. Abstract:
  5. We investigate singular stochastic delay differential equations (SDDEs) in view of their long-time behavior. Using Lyons's rough path theory, we show that SDDEs can be solved pathwise and induce a continuous stochastic flow on the space of (Gubinelli's) controlled paths. In the language of random dynamical systems, this result shows that SDDEs induce a continuous cocycle on random fibers, or, more precisely, on a measurable field of Banach spaces. We furthermore prove a multiplicative ergodic theorem (MET) on measurable fields of Banach spaces that applies under significantly weaker structural and measurability assumptions than preceding METs. Applying it to linear SDDEs shows that the induced cocycle possesses a discrete Lyapunov spectrum that can be used to describe the long-time behavior. © 2022 Society for Industrial and Applied Mathematics
  6. Keywords:
  7. Multiplicative ergodic theorem ; Random dynamical systems ; Rough paths ; Stochastic delay differential equation ; Differential equations ; Dynamical systems ; Lyapunov methods ; Stochastic systems ; Cocycles ; Dynamical theory ; Long time behavior ; Lyapunov spectrum ; Multiplicative ergodic theorem ; Random dynamical system ; Random fibers ; Rough path ; Stochastic delay differential equations ; Stochastic flows ; Banach spaces
  8. Source: SIAM Journal on Applied Dynamical Systems ; Volume 21, Issue 1 , 2022 , Pages 542-587 ; 15360040 (ISSN)
  9. URL: https://epubs.siam.org/doi/10.1137/21M1433435