Ornstein--Uhlenbeck Semigroups on Star Graphs

TL;DR

This paper establishes the existence and properties of Ornstein--Uhlenbeck semigroups on star graphs, including explicit formulas and invariant measures, extending classical results to this new setting.

Contribution

It introduces the first existence results for Ornstein--Uhlenbeck semigroups on star graphs with explicit formulas and invariant measures, extending classical theory.

Findings

Existence of classical solutions to parabolic problems on star graphs.

Explicit formulas for Ornstein--Uhlenbeck semigroups on star graphs.

Inheritance of regularity properties from classical Ornstein--Uhlenbeck semigroups.

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 $\mathbb R$.