Symmetry classes of classical stochastic processes

TL;DR

This paper classifies the symmetry properties of Markov generators in classical stochastic processes, revealing new spectral and dynamical features and constructing explicit examples for several symmetry classes.

Contribution

It extends the Bernard-LeClair symmetry classification to classical stochastic processes, identifying up to fifteen symmetry classes and constructing solutions for some classes.

Findings

Identification of up to fifteen symmetry classes for Markov generators.

Construction of explicit solutions for five classes with physical interpretation.

Discovery of spectral symmetries and time reversal properties in stochastic processes.

Abstract

We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions.