Free Markov processes and stochastic differential equations in von Neumann algebras

Abstract

Free Markov processes are investigated in Voiculescu’s free probability theory. We show that Voiculescu’s free Markov property implies a property called “weak Markov property”, which is the classical Markov property in the commutative case; while, in the general case, the “weak Markov property” is the same as the Markov property defined by Bozejko, Kummer, and Speicher. We also show that a kind of stochastic differential equations driven by free Levy processes has solutions. The solutions are free Markov processes.