Free Ornstein–Uhlenbeck processes

Free Ornstein–Uhlenbeck processes are studied in finite von Neumann algebras. It is shown that a free self-decomposable probability measure on R can be realized as the distribution of a stationary free Ornstein–Uhlenbeck process driven by a free Levy process. A characterization of a probability measure on R to be the stationary distribution of a periodic free Ornstein–Uhlenbeck process driven by a free Levy process is given in terms of the Levy measure of the measure. Finally, the notion of a free fractional Brownian motion is introduced. It is proved that the free stochastic differential equation driven by a fractional free Brownian motion has a unique solution. We call the solution a fractional free Ornstein–Uhlenbeck process.