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Ornstein–Uhlenbeck semigroups on star graphs

  • * Corresponding author: arhandi@unisa.it

    * Corresponding author: arhandi@unisa.it

Dedicated to the memory of Rosa Maria Mininni

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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  • 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} $.

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


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