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The local structure of \(q\)-Gaussian processes. (English) Zbl 1357.60081
Summary: The local structure of \(q\)-Ornstein-Uhlenbeck process and \(q\)-Brownian motion are investigated for all \(q \in(-1,1)\). These are classical Markov processes that arose from the study of noncommutative probability. These processes have discontinuous sample paths, and the local small jumps are characterized by tangent processes. It is shown that, for all \(q \in (-1,1)\), the tangent processes in the interior of the state space are scaled Cauchy processes possibly with drifts. The tangent processes at the boundary of the state space are also computed, but they are not well-known processes in classical probability theory. Instead, they can be associated with the free \(1/2\)-stable law, a well-known distribution in free probability, via Biane’s construction.

MSC:
60J60 Diffusion processes
60J65 Brownian motion
60G18 Self-similar stochastic processes
60G15 Gaussian processes
60G17 Sample path properties
60F17 Functional limit theorems; invariance principles
60J35 Transition functions, generators and resolvents
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