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 metric space \(E\) and characterized via a martingale problem for an operator \(A\). The authors obtain a criterion for invariant measures when \(A\) is just a subset of continuous functions on \(E\). The authors then also prove uniqueness of the solutions to a measure-valued evolution equation. For the case of locally compact state space, similar results were developed by others [cf. \textit{P. E. Echeverria}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 61, 1-16 (1982; Zbl 0476.60074); \textit{S. N. Ethier} and \textit{T. G. Kurtz}, ``Markov processes: Characterization and convergence'' (1986; Zbl 0592.60049), p. 252]. By applying authors' results to measure- valued filtering equation, uniqueness of the solutions to the analogue of the Zakai equation is proved in the class of all positive finite measures rather than only in a restricted class of measures.
uniqueness of the solutions to a measure-valued evolution equation
scientific article; zbMATH DE number 757767