Ornstein–Uhlenbeck semigroups on star graphs

Delio Mugnolo; Abdelaziz Rhandi

doi:10.3934/dcdss.2022030

Article Contents

2022, Volume 15, Issue 12: 3765-3779. Doi: 10.3934/dcdss.2022030

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Ornstein–Uhlenbeck semigroups on star graphs

Delio Mugnolo^1, and
Abdelaziz Rhandi^2, ,

1.
Lehrgebiet Analysis, Fakultät Mathematik und Informatik, Fern-Universität in Hagen, D-58084 Hagen, Germany
2.
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (Sa), Italy

^* Corresponding author: arhandi@unisa.it
^* Corresponding author: arhandi@unisa.it

Dedicated to the memory of Rosa Maria Mininni

Received: September 2021

Revised: December 2021

Early access: February 2022

Published: December 2022

The work of D.M. was supported by the Deutsche Forschungsgemeinschaft (Grant 397230547). A.R. is member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM). This work was started while A.R. visited the University of Hagen. He wishes to express his gratitude to the University of Hagen for the financial support. This article is based upon work from COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu

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Abstract

We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein–Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein–Uhlenbeck semigroup and give the unique associated invariant measure. We show that this semigroup inherits the regularity properties of the classical Ornstein–Uhlenbeck semigroup on $ \mathbb{R} $.

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Mathematics Subject Classification: 47D06, 34B45, 35J15, 35K10.

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