Non-commutative L\'evy processes for generalized (particularly anyon) statistics

Let $T=\mathbb R^d$. Let a function $Q:T^2\to\mathbb C$ satisfy $Q(s,t)=\bar{Q(t,s)}$ and $|Q(s,t)|=1$. A generalized statistics is described by creation operators $\partial_t^\dag$ and annihilation operators $\partial_t$, $t\in T$, which satisfy the $Q$-commutation relations. From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which $Q(s,t)$ is equal to $q$ if $s<t$, and to $\bar q$ if $s>t$. Here $q\in\mathbb C$, $|q|=1$. We start the paper with a detailed discussion of a $Q$-Fock space and operators $(\partial_t^\dag,\partial_t)_{t\in T}$ in it, which satisfy the $Q$-commutation relations. Next, we consider a noncommutative stochastic process (white noise) $\omega(t)=\partial_t^\dag+\partial_t+\lambda\partial_t^\dag\partial_t$, $t\in T$. Here $\lambda\in\mathbb R$ is a fixed parameter. The case $\lambda=0$ corresponds to a $Q$-analog of Brownian motion, while $\lambda\ne0$ corresponds to a (centered) $Q$-Poisson process. We study $Q$-Hermite ($Q$-Charlier respectively) polynomials of infinitely many noncommutatative variables $(\omega(t))_{t\in T}$. The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding L\'evy processes. To this end, we recursively define $Q$-cumulants of a field $(\xi(t))_{t\in T}$. This allows us to define a $Q$-L\'evy process as a field $(\xi(t))_{t\in T}$ whose values at different points of $T$ are $Q$-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a $Q$-L\'evy process, and derive a Nualart-Schoutens-type chaotic decomposition for such a process.