Distributions of Stable Processes on Spaces of Measurable Functions

Abstract

Let F = F(T,m) be a Banach function space of measurable functions on a σ finite measure space (T,m) and let K:F → L α be a kernel (an integral) operator into the space L α = L α(U, ω) of α-power integrable functions defined on another σ-finite measure space (U,ω). We consider a cylindrical probability μ defined by the characteristic function $$\hat{\mu}(f)=exp{-\int_{U}|Kf(u)|^{\alpha}w(du)}, \forall f\epsilon IF$$, for 0 < α < 2. This is the characteristic function of the probability distribution induced by a symmetric α-stable process given by an integral representation. We partly generalize integrability results of these processes by extending the cylindrical probability μ to a countably additive probability.