Infinite divisibility of sub-stable processes. Part I. geometry of sub-spaces of Lα-space

Generalizing the definition of sub-Gaussian processes we define a sub-stable process as a scale mixture of symmetric stable processes and study its infinite divisibility. This turns out to be strictly dependent on the geometry of a sub-space H(R) of the Lα-space generated by the corresponding stable process. This space plays a similar role as the reproducing kernel Hilbert space in the case of sub-Gaussian processes. We also investigate the uniqueness of the representation and some related questions in the language of geometrical properties of this space.