Ergodic properties of the nonlinear filter

In a recent work (Bhatt et al., SIAM J. Control Optim. 39 (2000) 928) various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a Feller–Markov process with values in a complete separable metric space E then the pair process (signal, filter) is also a Feller–Markov process with state space E×P(E), where P(E) is the space of probability measures on E. Furthermore, it was shown that if the signal has a unique invariant measure then, under appropriate conditions, uniqueness of the invariant measure for the above pair process holds within a certain restricted class of invariant measures. In many asymptotic problems concerning approximate filters (Budhiraja and Kushner, SIAM J. Control Optim. 37 (1997) 1946; 38 (2000) 1874) it is desirable to have the uniqueness of the invariant measure to hold in the class of all invariant measures. In this paper we first show that for a rich class of filtering problems, when the signal has a unique invariant measure, the property of “asymptotic stability” for the filter holds. Using this property of asymptotic stability we then provide sufficient conditions under which the (signal, filter) pair has a unique invariant measure. We also show that, in a certain sense, the property of asymptotic stability is necessary for the uniqueness of the invariant measure.