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On free stochastic processes and their derivatives

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 non-commutative $L^2$ of sample-space. We define a stochastic integral for our family of free processes.