On the Markov commutator

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-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 potentials which are sufficiently convex.

MSC

primary
60J10
secondary
15A27
20N20
52C99
39A12

Keywords

Symmetry group of a Markov kernel
Hypergroup property
Duality by intertwining
Birth and death chains
Metropolis algorithms
One-dimensional discrete wave equations
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