On Relations Between Urbanik and Mehler Semigroups

TL;DR

This paper explores the connection between Urbanik and Mehler semigroups in Banach spaces, showing how operator-selfdecomposable measures induce generalized Mehler semigroups and providing new proofs for their random integral representations.

Contribution

It establishes a link between Urbanik decomposability semigroups and generalized Mehler semigroups in Banach spaces, extending previous Hilbert space results and offering new proofs.

Findings

Operator-selfdecomposable measures induce generalized Mehler semigroups.

Generalized Mehler semigroups can be represented as random integrals of operator-valued functions.

New proofs of the random integral representation are provided.

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 L\'evy processes. Our Banach space setting is in the 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.