Title: Lost in Translation: Ergodic Theory and Stationary Processes in Correspondence
Speaker: Nachi Avraham Re'em (Technion)

Abstract:

Let G be a group. Giving a measure preserving action of G on a probability space together with a distinguished random variable, is the same as specifying a stationary process indexed by G. I will begin by describing a dictionary between two basic properties in these two perspectives:

1. The underlying probability space can be chosen to be standard corresponds precisely to what probabilists call separability in probability. This is a fairly elementary fact.

2. When G is a Polish group, the underlying action being jointly measurable corresponds exactly to what probabilists call stochastic continuity. This follows from a classical characterization of stochastic continuity due to Hoffmann-Jørgensen together with a fundamental observation of Glasner, Tsirelson, and Weiss.

I will then discuss how one can naturally take stochastically continuous extensions of stationary processes, a procedure of completing a separable in probability stationary process indexed by an abstract group into a stochastically continuous stationary process indexed by a Polish group. This basic idea is inspired by a suggestion of Sasha Danilenko in a correspondence with Emmanuel Roy and I. If time permits, I will explain how stochastically continuous extensions can be used to study ergodic properties.