Ornstein--Uhlenbeck semigroup on rooted trees

We study Ornstein--Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure.

Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root.

The associated semigroups are symmetric, analytic and positivity preserving; the Dirichlet semigroup is sub-Markovian, while the Neumann semigroup is Markovian and admits the Gaussian measure as its unique invariant measure up to scalar multiples.

We prove compactness of the resolvent and derive linear eigenvalue asymptotics.

For regular rooted trees, we adapt the Naimark--Solomyak decomposition to the Gaussian weighted setting, reducing the operators to one-dimensional half-line problems and obtaining refined spectral localisation and lower bounds.