Scaled Entropy of Filtrations of $\sigma$-Fields

Abstract

We study the notion of the scaled entropy of a filtration of $\sigma$-fields (i.e., decreasing sequence of $\sigma$-fields) introduced in [A. M. Vershik, Russian Math. Surveys, 55 (2000), pp. 677–733]. 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. 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 ${\bf Z}^d$) 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, we formulate the standardness criterion and describe its connections with the scaled entropy and the notion of a tower of measures.