Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

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For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.