A Dynamical Theory for Singular Stochastic Delay Differential Equations I: Linear Equations and a Multiplicative Ergodic Theorem on Fields of Banach Spaces
Mazyar Ghani Varzaneh, Sebastian Riedel, Michael Scheutzow
Abstract
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.