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

Abstract

We collect and prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck (OU) processes. Although processes considered in this paper were defined either in a noncommutative probability context or through quadratic harnesses we define them once more as a “continuous time” generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say ${\bf X=}( X_{t})_{t\ge 0}$ called $q$-Wiener) depend on one parameter $q\in(-1,1]$, and those of the second one (say ${\bf Y=}(Y_{t})_{t\in{\bf R}}$ called $(\alpha,q)$-Ornstein--Uhlenbeck) on two parameters $(\alpha,q)\in(0,\infty)\times(-1,1]$. The first one resembles the Wiener process in the sense that for $q=1$ it is a Wiener process but also that for $\vert q\vert <1$ and $\forall n\ge1$: $t^{n/2}H_{n}( X_{t}/\sqrt{t}\,|\,q),$ where $(H_{n})_{n\ge0}$ are the so-called $q$-Hermite polynomials, are martingales. However, it neither has independent increments not allows continuous sample path modification. The second one resembles the OU process. For $q=1$ it is a classical OU process. For $\vert q\vert <1$ it is also stationary with correlation function equal to $\exp(-\alpha|t-s|)$ and has many properties resembling those of its classical version. We think that these processes are fascinating objects to study posing many interesting, open questions.