Classical dilations à la Quantum Probability of Markov evolutions in discrete time

Details Classical dilations à la Quantum Probability of Markov evolutions in discrete time Classical dilations à la Quantum Probability of Markov evolutions in discrete time

2007-02-23

arxiv.org/abs/math/0702690v1

Mathematics

Probability

Scientific paper

We study the Classical Probability analogue of the 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 can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the 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.

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