On free stochastic processes and their derivatives

TL;DR

This paper investigates free stochastic processes with covariance kernels derived from tempered measures, introduces an orthonormal basis in non-commutative $L^2$, and defines a stochastic integral for these processes, advancing free probability analysis.

Contribution

It introduces a new family of free stochastic processes with covariance kernels from tempered measures and constructs an orthonormal basis and stochastic integral in this framework.

Findings

Established orthonormal bases in non-commutative $L^2$ spaces.

Defined stochastic integrals for free processes.

Connected free processes with semi-circle distributions.

Abstract

We study a family of free stochastic processes whose covariance kernels $K$ may be derived as a transform of a tempered measure $\sigma$. 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 non-commutative $L^2$ of sample-space. We define a stochastic integral for our family of free processes.