Ornstein--Uhlenbeck Semigroups on Star Graphs

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 inherit the regularity properties of the classical Ornstein–Uhlenbeck semigroup on R.

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