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arXiv · 1511.06667

The local structure of $q$-Gaussian processes

Włodzimierz Bryc·Yizao Wang
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Abstract

The local structure of $q$-Ornstein-Uhlenbeck processes and $q$-Brownian motions are investigated, for all $q\in(-1,1)$. These are the classical Markov processes corresponding to the noncommutative $q$-Gaussian processes. 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 at inner domain are scaled Cauchy processes possibly with drifts, and the tangent processes at the boundary of the domain are related to the free $1/2$-stable law via Biane's construction.

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Włodzimierz Bryc, Yizao Wang. 2016-02-25. The local structure of $q$-Gaussian processes. https://arxiv.org/abs/1511.06667

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