Relations Between Urbanik and Mehler Semigroups

Zbigniew Jurek, University of Wroclaw, Poland

Abstract. It is shown that operator-selfdecomposable measures, or more precisely their Urbanik decomposability semigroups, induce generalized Mehler semigroups of bounded linear operators. Moreover, those semigroups can be represented as random integrals of operator valued functions with respect to stochastic Lévy processes. Our Banach space setting is in contrast with that of Hilbert space, on which so far the generalized Mehler semigroups were studied.