## The Annals of Probability

### Conditional ergodicity in infinite dimension

#### Abstract

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. Föllmer, 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.

#### Article information

Source
Ann. Probab. Volume 42, Number 6 (2014), 2243-2313.

Dates
First available in Project Euclid: 30 September 2014

https://projecteuclid.org/euclid.aop/1412083626

Digital Object Identifier
doi:10.1214/13-AOP879

Mathematical Reviews number (MathSciNet)
MR3265168

Zentralblatt MATH identifier
1351.37032

#### Citation

Tong, Xin Thomson; van Handel, Ramon. Conditional ergodicity in infinite dimension. Ann. Probab. 42 (2014), no. 6, 2243--2313. doi:10.1214/13-AOP879. https://projecteuclid.org/euclid.aop/1412083626

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