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Summary

Markov and Feller semigroups are introduced, together with the corresponding stochastic processes. As all generators of Feller semigroups satisfy the positive maximum principle, we focus on that property and discuss the associated Hille–Yosida–Ray theorem. The main result of the chapter is proof of the Courrege theorem, which gives a Levy–Khinchine representation (but with variable coefficients) for all linear operators satisfying the positive maximum principle. We conclude with a brief discussion of the martingale problem and sub-Feller semigroups.

We consider a system of particles in birth and death on a stochastic flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of bounded counting measures. We show that its transition semigroup is a Feller semigroup and exhibit its pregenerator. The pregenerator defines a martingale problem. We show that the particle process solves the problem uniquely.

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7 - Markov and Feller Semigroups

Summary

Birth and death on a Brownian flow: a Feller semigroup and its generator and a martingale problem

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7 - Markov and Feller Semigroups

Summary

Birth and death on a Brownian flow: a Feller semigroup and its generator and a martingale problem