Geodesic
Dilations \`a la Quantum Probability of Markov Evolutions in Discrete Time
Teoriâ veroâtnostej i ee primeneniâ, Tome 54 (2009) no. 1, pp. 191-201

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We study the classical probability analogue of the unitary dilations of a quantum dynamical semigroup in quantum probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space $E$, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system $E$ with this environment such that the original Markov evolution of $E$ is realized by a proper choice of the initial random state of the environment. We also compare these dilations with the unitary dilations of a quantum dynamical semigroup in quantum probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of $*$-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.
Keywords: quantum dynamical semigroup, unitary dilation.
Mots-clés : Markov chain 
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     author = {M. Gregoratti},
     title = {Dilations \`a la {Quantum} {Probability} of {Markov} {Evolutions} in {Discrete} {Time}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     url = {http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a12/}
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M. Gregoratti. Dilations \`a la Quantum Probability of Markov Evolutions in Discrete Time. Teoriâ veroâtnostej i ee primeneniâ, Tome 54 (2009) no. 1, pp. 191-201. http://geodesic.mathdoc.fr/item/TVP_2009_54_1_a12/