Marjeta Kramar Fijavž: Ornstein-Uhlenbeck operators on rooted trees

Ornstein–Uhlenbeck operators and semigroups appear in several fields from quantum mechanics to stochastic analysis. We will first review some general properties of these objects in the simplest one-dimensional case. Then we shall focus to the Ornstein–Uhlenbeck operators acting along the edges of a rooted metric tree equipped with a Gaussian-type measure. Extending known results on star graphs, we shall construct two self-adjoint realisations that generate symmetric, analytic Markov semigroups. Both operators have compact resolvent and pure point spectrum with at least linear eigenvalue growth. For regular trees, we can further provide a decomposition into one-dimensional Ornstein–Uhlenbeck operators along the half-lines, leading to refined spectral estimates.

Based on joint work in progress with Sahiba Arora (Hannover), Delio Mugnolo (Hagen), and Abdelaziz Rhandi (Salerno).