Classical dilations \`a la Hudson-Parthasarathy of Markov semigroups
Abstract
We study the Classical Probability analogue of the 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 dilations of a quantum dynamical semigroup in Quantum Probability: given a classical Markov semigroup, we extend it to a proper quantum dynamical semigroup for which we can find a Hudson-Parthasarathy dilation which is itself an extension of our classical dilation.
Cite
@article{arxiv.math/0702784,
title = {Classical dilations \`a la Hudson-Parthasarathy of Markov semigroups},
author = {M. Gregoratti},
journal= {arXiv preprint arXiv:math/0702784},
year = {2008}
}