$q$-Wiener and $(\alpha,q)$-Ornstein--Uhlenbeck Processes. A Generalization of Known Processes

P. J. Szabł owski
2012 Theory of Probability and its Applications  
We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two
more » ... onal moments. The finite dimensional distributions of the first one (say X=(X_t)_t≥0 called q-Wiener) depends on one parameter q∈(-1,1] and of the second one (say Y=(Y_t)_t called (α,q)- Ornstein--Uhlenbeck) on two parameters (α,q)∈(0,∞)×(-1,1]. The first one resembles Wiener process in the sense that for q=1 it is Wiener process but also that for |q|<1 and ≥1: t^n/2H_n(X_t/|q), where (H_n)_n≥0 are the so called q-Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU process. For |q|<1 it is also stationary with correlation function equal to exp(-α|t-s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.
doi:10.1137/s0040585x97985674 fatcat:yfrjzpp6xjcrhndvlv6d6xfy3q