Asymptotic behavior of the nonlinear filtering errors of Markov processes

Nonlinear filtering process [pi]t, t >= 0 of a Markovian signal process with the state space S is regarded as a stochastic process with values in the set of all probability distributions over S. Under a suitable condition, it is shown that the filtering process is Markovian and that the invariant measure of the filtering process exists uniquely if and only if the stationary signal process (flow) is purely nondeterministic. These results are applied to the study for the asymptotic behavior of the filtering error. It turns out that the minimal asymptotic error is 0 if the signal process is transient, null recurrent or deterministic positive recurrent.