EVOLUTION-EQUATIONS FOR MARKOV-PROCESSES - APPLICATION TO THE WHITE-NOISE THEORY OF FILTERING

Let X be a Markov process taking values in a complete, separable metri c space E and characterized via a martingale problem for an operator A . We develop a criterion for invariant measures when range A is a subs et of continuous functions on E. Using this, uniqueness in the class o f all positive finite measures of solutions to a (perturbed) measure-v alued evolution equation is proved when the test functions are taken f rom the domain of A. As a consequence, it is shown that in the charact erization of the optimal filter (in the white-noise theory of filterin g) as the unique solution to an analogue of Zakai (as well as Fujisaki -Kallianpur-Kunita) equation, it suffices to take domain A as the clas s of test functions where the signal process is the solution to the ma rtingale problem for A.