An ergodic process of zero divergence-distance from its time-reversed process

TL;DR

This paper constructs an ergodic process with zero divergence-distance from its time-reversal, demonstrating the existence of time-asymmetric processes indistinguishable under reversal and potentially generated with zero dissipation.

Contribution

It introduces a novel ergodic process with zero divergence-distance from its time-reversal, challenging assumptions about time-symmetry and dissipation in stochastic processes.

Findings

Existence of time-asymmetric ergodic processes with zero divergence-distance

Construction of such processes using universal coding techniques

Implication that some processes can be time-reversal indistinguishable with zero dissipation

Abstract

An ergodic process $P$ is constructed such that the divergence-rate $D(P || P^*)$ is zero, yet $P$ is not equal to its time-reversed process $P^*$. The process $P$ is constructed as a special realization of the universal coding found by Xu. This result shows that there exist time-asymmetric processes that are indistinguishable under time-reversal and that can, in principle, be generated with zero dissipation.