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Shifted generalized Mehler semigroups on white noise functionals

Cig Un, Lee Mi Ra

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

185--205

We study shifted generalized Mehler semigroups on white noise functionals. We prove characterizations of invariant (white noise) distribution and the covariance property for shifted generalized Mehler semigroups. Also, we prove a Liouville type property of a shifted generalized Mehler semigroup or its infinitesimal generator.

On relations between Urbanik and Mehler semigroups

Jurek Z. J.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

297--308

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.