On the Markov commutator

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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-Trimèche'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-Trimèche theorem, birth and death chains, Metropolis algorithms, one-dimensional discrete wave equations.

Mots clés: one-dimensional discrete wave equations; Metropolis algorithms; birth and death chains; Achour-Trimèche theorem; finite Markov kernels; duality by intertwining; hypergroup property; Markov commutator; symmetry group of a Markov kernel

Domaines: Probabilités [math.PR]

Laurent Miclo. On the Markov commutator. Bulletin des Sciences Mathématiques, 2019. ⟨hal-01143511v2⟩