Continuous flows driving Markov processes and multiplicative $L^p$-semigroups

TL;DR

This paper introduces a method to drive Markov processes using continuous flows, establishing conditions for the continuity of these flows and their relation to multiplicative semigroups on Lp spaces, with applications to measure-valued superprocesses.

Contribution

It develops a framework connecting continuous flows with Markov processes and multiplicative semigroups, extending weak generators and martingale problems to unbounded functions.

Findings

Any flow is continuous in a suitable topology.

Markovian multiplicative semigroups on Lp are generated by continuous flows.

Extension of weak generator and martingale problem to unbounded functions.

Abstract

We develop a method of driving a Markov processes through a continuous flow. In particular, at the level of the transition functions we investigate an approach of adding a first order operator to the generator of a Markov process, when the two generators commute. A relevant example is a measure-valued superprocess having a continuous flow as spatial motion and a branching mechanism which does not depend on the spatial variable. We prove that any flow is actually continuous in a convenient topology and we show that a Markovian multiplicative semigroup on an Lp space is generated by a continuous flow, completing the answer to the question whether it is enough to have a measurable structure, like a C0-semigroup of Markovian contractions on an $L^p$-space with no fixed topology, in order to ensure the existence of a right Markov process associated to the given semigroup. We extend from bounded to unbounded functions the weak generator (in the sense of Dynkin) and the corresponding martingale problem