Filter Stability in Infinite Dimensional Systems

Abstract

Filtering is a process that sequentially assimilates data from observations and generates a probabilistic description of a hidden underlying process. The goal of this thesis is to study the ergodic behavior of the filtering process in infinite dimensional systems. Such a framework brings recent developments in infinite dimensional systems into the filtering scenario. We approach this goal by introducing an ergodic property named local ergodicity, which generalizes a notion of H. Follmer. This property interacts well with the conditioning structure and hence can be inherited through filtering. We proceed to connect local ergodicity with a topological method developed for infinite dimensional systems named asymptotic coupling. Finally, we show how to apply our framework to fluid models in the forms of stochastic Navier Stokes equations.