Generalized Mehler semigroups:: The non-Gaussian case

We study generalized Mehler semigroups, introduced in [7], with special emphasis on the non-Gaussian case. We review and simplify the method of construction. In the general (non-Gaussian) case we construct an associated cadlag Markov process in an appropriate state space obtained as a solution of a stochastic equation which can be solved 'omega by omega'. We also show tightness of the associated (r, p)-capacities. Invariant measures, time regularity and a definition of the generator are also studied.