Measure-valued Markov processes and stochastic flows on abstract spaces

Abstract

We consider measure-valued processes with constant mass in Hilbert space. The stochastic flow which carries the mass satisfies a stochastic differential equation with coefficients depending on the mass distribution. This mass distribution can be considered as the conditional distribution of the solution of a certain SDE. In contrast to the filtration equation, in our case the random measure cannot diffuse: a single particle cannot break up or turn into clouds. The Markov structure of the measure-valued processes obtained is studied and a comparison with Fleming–Viot processes is presented.