Scaled entropy of filtrations of $\sigma$-fields

TL;DR

This paper introduces a method to compute the scaled entropy of filtrations of $\sigma$-fields associated with Markov processes driven by random walks on Bernoulli actions of groups, revealing invariants that distinguish different group dimensions and Bernoulli entropies.

Contribution

It provides a new computational approach for scaled entropy in filtrations of $\sigma$-fields related to Markov processes on group actions, and demonstrates its invariance properties.

Findings

Scaled entropy distinguishes filtrations of different group dimensions.

Sequences of $\sigma$-fields are metrically nonisomorphic for different dimensions.

Scaled entropy varies with Bernoulli scheme entropy.

Abstract

We study the notion of the scaled entropy of a filtration of $\sigma$-fields (= decreasing sequence of $\sigma$-fields) introduced by the first author ({V4}). We suggest a method for computing this entropy for the sequence of $\sigma$-fields of pasts of a Markov process determined by a random walk over the trajectories of a Bernoulli action of a commutative or nilpotent countable group (Theorems~5,~6). Since the scaled entropy is a metric invariant of the filtration, it follows that the sequences of $\sigma$-fields of pasts of random walks over the trajectories of Bernoulli actions of lattices (groups ${\Bbb Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the same $d$ but different values of the entropy of the Bernoulli scheme. We give a brief survey of the metric theory of filtrations, in particular, formulate the standardness criterion and describe its connections with the scaled entropy and the notion of a tower of measures.