Consider a bivariate ergodic Markov process. One component is observed and the other component is hidden. What can we say about the ergodic properties of the hidden process conditionally on the observed process? Such questions are of direct relevance to the performance of nonlinear filtering algorithms that are widely used by engineers. Mathematically, the question hinges on an insidious measure-theoretic problem that appears in many probabilistic settings and remains far from well understood. When the underlying Markov process is ergodic in total variation, the conditional ergodic properties are inherited under mild assumptions (this resolves a long-standing problem in this area). Under weaker notions of ergodicity, however, new phenomena can appear. Things get even worse when one considers random fields rather than Markov processes. I will discuss these results, and describe some ongoing work aimed at understanding conditional ergodic properties in high dimensional settings.
Speaker: Ramon van Handel, Princeton University