On Relations Between Urbanik and Mehler Semigroups

Details On Relations Between Urbanik and Mehler Semigroups On Relations Between Urbanik and Mehler Semigroups

2008-11-18

arxiv.org/abs/0811.2989v1

Probability and Mathematical Statistics vol.29 (2009), pp. 297-308

Mathematics

Probability

14 pages

Scientific paper

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.

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