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 -