On free stochastic processes and their derivatives

Abstract

Abstract. We study a family of free stochastic processes whose covariance kernels K may be derived as a transform of a tempered measure σ. These processes arise, for example, in consideration non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal bases in the corresponding noncommutative L 2 of sample-space. We define a stochastic integral for our family of free processes.

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

“…The literature on Gaussian Hilbert space, white noise analysis, and its relevance to Malliavin calculus is vast; we limit ourselves here to citing [17,[36][37][38][39][40][41], and the papers cited there.…”

Free W*-Dynamical Systems From p-Adic Number Fields and the Euler Totient Function

“…The Euler totient function φ is so famous, important, and applicable in both classical and modern number theory that we cannot help emphasizing the importance of this function not only in mathematics but also in other scientific areas (e.g., [4,5,22,23,29]). …”

“…The free probability theory was pioneered by D. Voiculescu (e.g., [1,2]) and motivated by a question in von Neumann algebra (alias W * -algebra) theory, the free-group factors isomorphism problem (e.g., [2,3]). There has been a recent renewed interest in analysis on free probability spaces, especially in connection with free random processes (e.g., [4,5]) In this paper, we consider connections between the two independent free-probabilistic models induced from number-theoretic objects, (i) free probability spaces (M p , ϕ p ) of the von Neumann algebras M p generated by p-adic number fields Q p and the corresponding integrations ϕ p on M p , (e.g., [6][7][8]) and (ii) free probability spaces (A, g p ) of the algebra A consisting of all arithmetic functions, equipped with the usual functional addition (+) and the convolution ( * ), and the point-evaluation linear functionals g p on A, for all primes p (e.g., [9][10][11][12]). And we apply such relations to study W * -dynamical systems induced by Q p (e.g., [9]).…”

Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra

“…To obtain the socalled full Fock space, one includes the antisymmetric Fock space, which can be associated with fermions. In this context, i.e., when considering the full Fock space, it is also possible to define a noncommutative analog of the Kondratiev space [5,8]. Moreover, the same type of tools can be developed in the framework of Q-deformed commutation relations [25].…”