ON RELATIONS BETWEEN URBANIK AND MEHLER SEMIGROUPS
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 the Hilbert spaces on which so far and most often the generalized Mehler
semigroups were studied. Furthermore, we give new proofs of the random integral
representation.
2000 AMS Mathematics Subject Classification: Primary: 60B12, 60E07; Secondary:
47D06, 46G10, 47D60.
Keywords and phrases: Banach space; one-parameter strongly continuous semigroup;
Urbanik decomposability semigroup; measure valued cocycle; generalized Mehler
semigroup; Lévy process; random integral.