On the Markov commutator

TL;DR

This paper investigates the properties of the Markov commutator associated with finite Markov kernels, exploring its relation to hypergroups, duality, and wave equations, and demonstrating limitations of certain preservation properties.

Contribution

It introduces new insights into the structure of the Markov commutator and extends the understanding of the Achour-Trim{\

Findings

The discrete analogue of the Achour-Trim{\

Preservation of non-negativity holds for symmetric and monotone convex potentials.

The Markov commutator relates to hypergroup properties and duality by intertwining in finite Markov kernels.

Abstract

The Markov commutator associated to a finite Markov kernel P is the convex semigroup consisting of all Markov kernels commuting with P. Its interest comes from its relation with the hypergroup property and with the notion of Markovian duality by intertwining. In particular, it is shown that the discrete analogue of the Achour-Trim{\`e}che's theorem, asserting the preservation of non-negativity by the wave equations associated to certain Metropolis birth and death transition kernels, cannot be extended to all convex potentials. But it remains true for symmetric and monotone convex potentials. Keywords: finite Markov kernels, Markov commutator, symmetry group of a Markov kernel, hypergroup property, duality by intertwining, Achour-Trim{\`e}che theorem, birth and death chains, Metropolis algorithms, one-dimensional discrete wave equations.