Some remarks on spectral representation for symmetric stable processes

Abstract

For the symmetric α-stable stochastic process X={Xt∶t∈T} with reproducing kernel space H(X) ⊂ Lα constructed in § 1 we define the following parameters: $$\alpha _0 = \sup {\mathbf{ }}\{ \beta \in (0.2]:{\mathbf{ }}\mathcal{H}\mathcal{X}$$ embeds isometrically into some Lβ}, containsl β 's uniformly}. In §2 we show that for α0 > α the stochastic process X admits the representation $$X_t = \smallint Y_t (w){\mathbf{ }}Z_\alpha (dw),{\mathbf{ }}t \in T,$$ where {Yt∶t∈T} itself is a symmetric stable process and Zα is a symmetric α-stable independently scattered random measure. We show also how some properties of the stochastic process {Xt∶t∈T} depend on the corresponding properties of the process {Yt∶t∈T}.