The goal of this paper is to develop a general method to establish
conditional ergodicity of infinite-dimensional Markov chains. Given a Markov
chain in a product space, we aim to understand the ergodic properties of its
conditional distributions given one of the components. Such questions play a
fundamental role in the ergodic theory of nonlinear filters. In the setting of
Harris chains, conditional ergodicity has been established under general
nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional
state spaces are rarely amenable to the classical theory of Harris chains due
to the singularity of their transition probabilities, while topological and
functional methods that have been developed in the ergodic theory of
infinite-dimensional Markov chains are not well suited to the investigation of
conditional distributions. We must therefore develop new measure-theoretic
tools in the ergodic theory of Markov chains that enable the investigation of
conditional ergodicity for infinite dimensional or weak-* ergodic processes. To
this end, we first develop local counterparts of zero-two laws that arise in
the theory of Harris chains. These results give rise to ergodic theorems for
Markov chains that admit asymptotic couplings or that are locally mixing in the
sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary
absolutely regular sequences. We proceed to show that local ergodicity is
inherited by conditioning on a nondegenerate observation process. This is used
to prove stability and unique ergodicity of the nonlinear filter. Finally, we
show that our abstract results can be applied to infinite-dimensional Markov
processes that arise in several settings, including dissipative stochastic
partial differential equations, stochastic spin systems and stochastic
differential delay equations.Comment: Published in at http://dx.doi.org/10.1214/13-AOP879 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
