Evolution equations for Markov processes: application to the white-noise theory of filtering

Abstract

Let X be a Markov process taking values in a complete, separable metric spaceE and characterized via a martingale problem for an operator A. We develop a criterion for invariant measures when range A is a subset of continuous functions on E. Using this, uniqueness in the class of all positive finite measures of solutions to a (perturbed) measure-valued evolution equation is proved when the test functions are taken from the domain of A. As a consequence, it is shown that in the characterization of the optimal filter (in the white-noise theory of filtering) as the unique solution to an analogue of Zakai (as well as Fujisaki-Kallianpur-Kunita) equation, it suffices to take domain A as the class of test functions where the signal process is the solution to the martingale problem for A.