Ergodicity of the finite and infinite dimensional alpha-stable systems

Some finite and infinite dimensional perturbed alpha-stable dynamics are constructed and studied in this paper. We prove that the finite dimensional system is strongly mixing, while in the infinite dimensional case that the functional coercive inequal- ities are not available, we develop and apply a technique to prove the point-wise ergodicity for systems with suffciently small interaction in a large subspace of omega= RZd .