Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups
type: Item
instance of: scholarly article
title: Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups
zbMATH Open document ID: 1242.60006
author: Wilfried Hazod
published in: Semigroup Forum
publication date: 2012-03-23T00:00:00Z
full work available at URL: http://hdl.handle.net/2003/25966
review text: 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.
reviewed by: Michael Voit
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 60G18
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 6018265
zbMATH Keywords: Mehler semigroups
zbMATH Keywords: Ornstein-Uhlenbeck processes
zbMATH Keywords: Bessel hypergroups on matrix cones
zbMATH Keywords: background driving Levy processes
MaRDI profile type: Publication
OpenAlex ID: W2039011522
cites work: Stable and semistable hemigroups: Domains of attraction and self-decomposability
cites work: Random integral representation of operator-semi-self-similar processes with independent incre\-ments.
cites work: Q3622167
cites work: Q5292403
cites work: Generalized Mehler semigroups and applications
cites work: Harmonic analysis of probability measures on hypergroups
cites work: Note on product formulas for operator semigroups
cites work: Product formulas, nonlinear semigroups, and addition of unbounded operators
cites work: Analytische Vektoren von Faltungshalbgruppen. I
cites work: Generalized Mehler semigroups and catalytic branching processes with immigration
cites work: Représentations de semi-groupes de mesures sur un groupe localement compact
cites work: Semi-groupes de mesures complexes et calcul symbolique sur les générateurs infinitesimaux de semi-groupes d'opérateurs
cites work: Semi-groupes d'opérateurs invariants et opérateurs dissipatifs invariants. (Semi-groups of invariant operators and invariant dissipative operators)
cites work: Q3721264
cites work: Stetige Faltungshalbgruppen von Wahrscheinlichkeitsmaßen und erzeugende Distributionen
cites work: Q4258048
cites work: Probability on Matrix-Cone Hypergroups: Limit Theorems and Structural Properties
cites work: Q2755106
cites work: Q4153376
cites work: Q3622169
cites work: Q5629957
cites work: Familles résolvantes, générateurs, cogénérateurs, potentiels
cites work: Q5657260
cites work: Some analytical semigroups occurring in probability theory
cites work: Self-similar processes with independent increments associated with Lévy and Bessel processes.
cites work: Q3969648
cites work: Q4385062
cites work: On Relations Between Urbanik and Mehler Semigroups
cites work: Q4863604
cites work: Q5526485
cites work: Q4357562
cites work: Skew Convolution Semigroups and Related Immigration Processes
cites work: Dissipative operators in a Banach space
cites work: Semi-Lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein-Uhlen\-beck type processes
cites work: On left invariant dissipative operators
cites work: Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues
cites work: Q3213733
cites work: Q5292412
cites work: Stochastic integrals in additive processes and application to semi-Lévy processes
cites work: Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type
cites work: On the equation \(\mu_{t+s} = \mu_s * T_s\mu_t\)
cites work: Lévy's probability measures on Euclidean spaces
cites work: Bessel convolutions on matrix cones: Algebraic properties and random walks
cites work: POSITIVE AND NEGATIVE DEFINITE FUNCTIONS ON THE DUAL SPACE OF A COMMUTATIVE HYPERGROUP
cites work: Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials
cites work: The Lévy-Khintchine-formula for dissipative distributions
DOI: 10.1007/S00233-011-9310-1
Recommended article: Q4523956
Recommended article: Probability on Matrix-Cone Hypergroups: Limit Theorems and Structural Properties
Recommended article: Q3221094
Recommended article: Q4833861
Recommended article: Bessel convolutions on matrix cones: Algebraic properties and random walks
Recommended article: Convolution semigroups and resolvent families of measures on hypergroups
Recommended article: Q4885798
Recommended article: Q4833870
Recommended article: The central limit theorem for stationary Markov processes with normal generator—with applications to hypergroups
Recommended article: Q3622169