Symmetry classes of classical stochastic processes

Sá, Lucas (author), Ribeiro, Pedro (author), Prosen, Tomaž (author), Bernard, Denis (author)

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.

• Language: English
• Keywords: statistical physics, stochastic processes
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 26 str.
• Numbering: Vol. 192, art. no. 41
• PID: 20.500.12556/RUL-175282
• UDC: 536
• ISSN on article: 1572-9613
• DOI: 10.1007/s10955-025-03423-y
• COBISS.SI-ID: 254405635
• Publication date in RUL: 23.10.2025

Record is a part of a journal

• Title: Journal of statistical physics : Elektronski vir
• Shortened title: J. stat. phys.
• Publisher: Kluwer
• ISSN: 1572-9613
• COBISS.SI-ID: 513205529

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: statistična fizika, stohastični procesi