Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups
Let \(G\) be a locally compact group or, more generally, a hypergroup, and \((T_t)_{t\in \mathbb R}\) a continuous one-parameter group of automorphisms of \(G\). Consider the associated semidirect product \(\Gamma:= G \rtimes\mathbb R\) . It is well-known for vector spaces \((G,+)\) that there is a one-to-one correspondence between Ornstein-Uhlenbeck type processes on \(G\) w.r.t. \((T_t)_{t\in \mathbb R}\) and associated so called background driving Levy processes on \(\Gamma\) on the level of processes and on the level of the associated operator semigroups. This kind of correspondence is also available for simply connected nilpotent Lie groups. In the paper under review, this correspondence is mainly extended to several commutative hypergroups \(G\), namely mainly the Bessel hypergroups on the cones \(\Pi_n\) of positive semidefinite matrices over the fields \(\mathbb R,\mathbb R, \mathbb H\) introduced by \textit{M. Rösler} [Compos. Math. 143, No. 3, 749--779 (2007; Zbl 1115.33011)]. These examples are of special interest as their theory is close to the vector groups \((\mathbb R,+)\), and as they admit a rich structure of nontrivial automorphism groups. The results rely on a detailed study of the cores of the generators of the associated semigroups.
scientific article; zbMATH DE number 6018265
scientific article