Mehler hemigroups and embedding of discrete skew convolution semigroups on simply connected nilpotent Lie groups

Abstract

It is shown how discrete skew convolution semigroups of probability measures on a simply connected nilpotent Lie group can be embedded into Lipschitz continuous semistable hemigroups by means of their generating functionals. These hemigroups are the distributions of increments of additive semi-selfsimilar processes. Considering these on an enlarged space-time group, we obtain Mehler hemigroups corresponding to periodically stationary processes of Ornstein-Uhlenbeck type, driven by certain additive processes with periodically stationary increments. The background driving processes are further represented by generalized Lie-Trotter formulas for convolutions, corresponding to a random integral approach known for finite-dimensional vector spaces.