On the Markov commutator

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{è}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{è}che theorem, birth and death chains, Metropolis algorithms, one-dimensional discrete wave equations.