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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
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 …