Details
Original language | English |
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Pages (from-to) | 542-587 |
Number of pages | 46 |
Journal | SIAM Journal on Applied Dynamical Systems |
Volume | 21 |
Issue number | 1 |
Early online date | 24 Feb 2022 |
Publication status | Published - Mar 2022 |
Abstract
Keywords
- multiplicative ergodic theorem, random dynamical systems, rough paths, stochastic delay differential equation
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Modelling and Simulation
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In: SIAM Journal on Applied Dynamical Systems, Vol. 21, No. 1, 03.2022, p. 542-587.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - A dynamical theory for singular stochastic delay differential equations I
T2 - linear equations and a multiplicative ergodic theorem on fields of Banach spaces
AU - Riedel, Sebastian
AU - Scheutzow, M.
AU - Ghani Varzaneh, M.
N1 - Funding Information: The work of the first author was supported by a scholarship from the Berlin Mathematical School(BMS). The work of the second and third authors was supported by the DFG via Research Unit FOR 2402.
PY - 2022/3
Y1 - 2022/3
N2 - We investigate singular stochastic delay differential equations (SDDEs) in view of their long-timebehavior. Using Lyons's rough path theory, we show that SDDEs can be solved pathwise and induce acontinuous stochastic flow on the space of (Gubinelli's) controlled paths. In the language of randomdynamical 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 multiplicativeergodic theorem (MET) on measurable fields of Banach spaces that applies under significantly weakerstructural and measurability assumptions than preceding METs. Applying it to linear SDDEs showsthat the induced cocycle possesses a discrete Lyapunov spectrum that can be used to describe thelong-time behavior.
AB - We investigate singular stochastic delay differential equations (SDDEs) in view of their long-timebehavior. Using Lyons's rough path theory, we show that SDDEs can be solved pathwise and induce acontinuous stochastic flow on the space of (Gubinelli's) controlled paths. In the language of randomdynamical 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 multiplicativeergodic theorem (MET) on measurable fields of Banach spaces that applies under significantly weakerstructural and measurability assumptions than preceding METs. Applying it to linear SDDEs showsthat the induced cocycle possesses a discrete Lyapunov spectrum that can be used to describe thelong-time behavior.
KW - multiplicative ergodic theorem
KW - random dynamical systems
KW - rough paths
KW - stochastic delay differential equation
UR - http://www.scopus.com/inward/record.url?scp=85127067916&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1903.01172
DO - 10.48550/arXiv.1903.01172
M3 - Article
VL - 21
SP - 542
EP - 587
JO - SIAM Journal on Applied Dynamical Systems
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
IS - 1
ER -