The perfection of local semi-flows and local random dynamical systems with applications to SDEs

TL;DR

This paper establishes a general perfection result for local semi-flows in Polish spaces, enabling their transformation into invariant local semi-flows and applications to stochastic differential equations driven by semimartingales.

Contribution

It introduces a novel perfection theorem for local semi-flows and demonstrates their application to SDEs with complex coefficients, previously unresolved.

Findings

Crude local semi-flows can be modified into perfect local semi-flows.

The results apply to SDEs driven by semimartingales with stationary increments.

Newly shows these SDEs generate local random dynamical systems.

Abstract

We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system. Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) random dynamical system or not.