Group invariant stable processes

We study symmetric α-stable (SαS) processes defined on a separable metric space T whose finite dimensional distributions are invariant un-der a group of transformations of T. This extends the classical notion of stationarity of stochastic processes. Minimal integral representation on standard Borel space S of SαS pro-cess which is group stationary (G-stationary) in the above sense corre-sponds to a group of isometries of L^α(S, μ). We show that for 1 < α < 2 this group of isometries is generated by a group action and a measur-able cocycle on S. The pair of group action and cocycle induced this way is unique up to an isomorphism of group actions and cohomology equivalence relation for cocycles. We show that if a group action ad-mits a Borel selector, then cocycles are cohomologous if and only if the induce the same homomorphisms on the isotropy groups. As an application we give characterization of isotropic SαS random fields on R² in terms of their minimal representations.