Classical Markov Processes from Quantum L

We show how classical Markov processes can be obtained from quantum L evy processes. It is shown that quantum L evy processes are quantum Markov processes, and suucient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without loosing the quantum Markov propertyyK um88]. Several examples, including the Az ema martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.