Dynamical Properties of Rough Delay Equations

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Ghani Varzaneh, Mazyar | 2022

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Type of Document: Ph.D. Dissertation
Language: Farsi
Document No: 55018 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Zohuri Zangeneh, Bijan
Abstract:
In this monograph, we investigate the long-time behavior of stochastic delay equations. Our approach is random dynamical systems, and we solve our equation in the rough path point of view. Namely, we deal with the singular case, i.e., when the delay terms also are appearing in the diffusion part. Although we can solve the equation using the classical tools of stochastic analysis, the main obstacle is the lack of flow property. More precisely, the solution does not depend continuously on the initial value. To solve this problem, we define this property differently. We will show how we can generate a flow property on fields of Banach spaces using rough path theory. As a consequence, we prove the cocycle property and establish a Wong-Zakai theorem. Since we use rough path theory, we can apply our results to the case where the noise consists of Brownian motions or fractional Brownian motions with 1/3
Keywords:
Invariant Manifolds ; Random Dynamical System ; Rough Paths ; Delay Equations ; Multiplicative Ergodic Theorem ; Dynamic Behavior

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