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

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