# A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces

## TL;DR

This paper develops a theoretical framework for singular stochastic delay differential equations, establishing a Multiplicative Ergodic Theorem on Banach space fields and applying it to linear cases, with implications for stability analysis.

## Contribution

It introduces a novel Multiplicative Ergodic Theorem for fields of Banach spaces and applies it to linear singular SDDEs, advancing the mathematical understanding of these equations.

## Key findings

- Established a Multiplicative Ergodic Theorem on Banach space fields
- Applied the theorem to analyze linear singular SDDEs
- Provided groundwork for stability analysis of non-linear singular SDDEs

## Abstract

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.