Dr. Ramon van Handel (ORFE, Princeton)

Title:  The ergodic theory of nonlinear filters

Abstract: The goal of nonlinear filtering is to estimate a process of interest given noisy and incomplete observations. Nonlinear filtering methods play a role in a wide variety of applications, ranging from robotics to finance. In many applications, one is interested in understanding the performance of the nonlinear filter (as well as related Monte Carlo algorithms or sequential decision procedures) over a long time horizon. Mathematically, this requires an understanding of the ergodic theory of nonlinear filters. Such a theory was first developed in a classic paper of H. Kunita (1971). Unfortunately, the key part of the proof in this paper contains a fundamental measure-theoretic error, which lies at the heart of the ergodicity problem for nonlinear filters.

In this talk, I will discuss recent progress in understanding the general ergodic theory of nonlinear filters. I will introduce the central measure-theoretic identity and outline its proof under very general assumptions, by means of the theory of Markov chains in random environments. I will also discuss two surprising counterexamples where the filter fails to be ergodic. The upshot is that the ergodicity of classical nonlinear filtering problems is now largely resolved, but nonlinear filtering problems with infinite dimensional signals (such as appear in applications to weather prediction or data assimilation) remain a mystery.