bibkey: saribeiroprosenbernard2024symmetryclasses
authors: Lucas Sá; Pedro Ribeiro; Tomaž Prosen; Denis Bernard
year: 2025
title: "Symmetry Classes of Classical Stochastic Processes"
doi: 10.1007/s10955-025-03423-y
url: https://arxiv.org/abs/2406.17955v2
claim: "A Markov generator is a real matrix with nonnegative off-diagonal entries and zero column sums (Eqs. (2)–(4)). For a generator with a commuting symmetry, the authors conjecture that the irreducible block containing the invariant measure can be recast into Markov form (§2, after Eq. (5)), and prove this under assumption (A3) (Appendix A)."
strata_touched:
  - D5/S3/StatisticalMechanics/SteadyStateBlockMarkovFormRefutation
license: citation-only
triage: anchor

Sá, Ribeiro, Prosen and Bernard, symmetry classes of classical stochastic processes

L. Sá, P. Ribeiro, T. Prosen and D. Bernard, Symmetry Classes of Classical Stochastic Processes, J. Stat. Phys. 192, 41 (2025), DOI 10.1007/s10955-025-03423-y; arXiv:2406.17955 [cond-mat.stat-mech, cross-listed math-ph] (v1 25 June 2024; v2 12 March 2025).

Verified locator

DOI: 10.1007/s10955-025-03423-y (Crossref: Journal of Statistical Physics 192, article 41, published 12 March 2025). Primary version read: https://arxiv.org/abs/2406.17955v2; the text was read from the PDF.

• §1: Eqs. (1)–(4), the Markov equation and the three properties of a Markov generator;
• §2: Eq. (5), irreducible blocks of a commuting symmetry, and the conjecture after it;
• Appendix A: the proof under assumption (A3).

The journal typesetting was not compared line by line.

Source statements

§1: “Conservation of probability requires that the Markov generator, an N × N matrix L, satisfies the three properties: L_ij ∈ R, (2) L_ij ≥ 0, i ≠ j, (3) Σ_{i=1}^N L_ij = 0, ∀j. (4)”

§2: “If L has a commuting symmetry, U L U^{−1} = L, (5) then, L and U can be simultaneously diagonalized. Inside each block of L, U acts trivially as a multiple of the identity. If no further commuting symmetries exist, we call such a block an irreducible block. … It is, however, natural to conjecture that the block containing the invariant measure of the Markov process (henceforth referred to as the steady-state block) can still be recast into Markov form. Indeed, consider an initial probability vector fully supported in the steady-state block. Under the Markov dynamics, the initial probability vector evolves into another admissible probability vector. … We conclude that the steady-state block maps probability vectors into probability vectors and it should be possible to write it in Markov form.”

Appendix A: “To proceed, we assume that the steady-state block corresponds to the symmetric sector of U and, furthermore, that p_{a|i} ∈ {0, 1}. (A3)”

Reading used by the repository

The symmetry U is real orthogonal and commutes with L; the steady-state block is the column space of an orthonormal real matrix V that is an entire eigenspace of U, is invariant under L and contains a stationary probability. It is irreducible when every complex unitary commuting with the restriction A = VᵀLV is a scalar. “Recast into Markov form” is read as: some invertible complex matrix S makes S⁻¹AS a real matrix satisfying Eqs. (2)–(4).