Dilations à la Quantum Probability of Markov Evolutions in Discrete Time

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.