Entropy of probability kernels from the backwards tail boundary

TL;DR

This paper proves that the operator entropy of Markov kernels equals the classical Kolmogorov-Sinai entropy for all cases, unifying previous partial results and clarifying the relationship between these entropy concepts.

Contribution

It establishes the equality of operator entropy and Kolmogorov-Sinai entropy universally, resolving a longstanding question in the theory of entropy for Markov operators.

Findings

Operator entropy equals Kolmogorov-Sinai entropy in all cases.

Unifies previous partial results on entropy equivalence.

Clarifies the relationship between Markov operators and classical dynamical systems.

Abstract

A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space $(X,\mu)$. These have culminated in a proof by Downarovicz and Frej that these definitions all coincide, and that the resulting quantity is uniquely characterized by certain properties. On the other hand, Makarov has shown that this `operator entropy' is always dominated by the Kolmogorov-Sinai entropy of a classical system that may be constructed from a Markov operator, and that these numbers coincide under certain extra assumptions. This note proves that equality in all cases.