Dilations à la Hudson–Parthasarathy of Markov Semigroups in Classical Probability

Abstract

We study the Classical Probability analogue of the unitary dilations of a quantum dynamical semigroup defined in Quantum Probability via quantum stochastic differential equations. Given a homogeneous Markov chain in continuous 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 unitary dilations of a quantum dynamical semigroup in Quantum Probability.

Keywords:

• Markov semigroup
• Quantum dynamical semigroup
• Unitary dilation

Mathematics Subject Classification:

• 60J27
• 81S25
• 46L53