Ornstein--Uhlenbeck semigroup on rooted trees

Abstract: 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.

• The paper develops a detailed functional analytic framework for Ornstein–Uhlenbeck operators on rooted trees, establishing self-adjoint Dirichlet and Neumann realizations with discrete spectra.
• The paper employs orthogonal decomposition leveraging radial symmetry to reduce the spectral problem to one-dimensional OU problems, yielding sharp spectral localization and precise eigenvalue estimates.
• The paper derives explicit linear spectral asymptotics, demonstrating that eigenvalues scale with the number of infinite branches and refining classical Naimark–Solomyak results.

Ornstein--Uhlenbeck Semigroup on Rooted Trees: Operator Theory, Spectral Asymptotics, and Structural Decomposition

Introduction

The study develops a detailed functional analytic and spectral theory for Ornstein–Uhlenbeck (OU) operators on rooted metric trees equipped with a Gaussian-type measure. Expanding on prior analyses of star graphs [Mugnolo & Rhandi, DCDS-S, 2022], this work rigorously constructs Dirichlet and Neumann realisations of the OU operator on much more general rooted trees, analyzes semigroup properties, and establishes leading-order spectral asymptotics. For regular rooted trees, the Naimark–Solomyak decomposition is refined to the Gaussian-weighted setting, enabling sharp spectral localisation and minimum eigenvalue estimates.

Operator Construction on Rooted Metric Trees

Consider a rooted metric tree $T$, consisting of a compact core $\widetilde{T}$ and finitely many half-lines attached to boundary vertices. Each edge is endowed with an arc-length coordinate, and the geometry induces a unique radial coordinate from the root. The Gaussian-type measure $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$ (depending only on radial distance) is imposed. Edge-wise Sobolev and $L^2_\mu$ spaces are defined, with continuity and Kirchhoff-type transmission conditions across vertices.

The Ornstein–Uhlenbeck operator is given by

$(A f)(x) = -\frac{1}{2} f''(x) + x f'(x),$

acting edgewise, with realisations distinguished by boundary conditions at the root: Dirichlet (killing at the root) and Neumann (reflection/conservative).

Dirichlet and Neumann realisations are constructed as self-adjoint operators on $L^2_\mu(T)$ via closed, symmetric, and strongly local Dirichlet forms. Compactness of the embedding of the form domain into $L^2_\mu(T)$ is established, implying a discrete spectrum for both operators.

Semigroup Properties and Invariant Measure

Both realisations generate contractive, analytic, compact $C_0$-semigroups on $L^2_\mu(T)$. The Neumann semigroup is Markovian and preserves positivity, admitting the Gaussian measure as its unique invariant measure (up to constant multiples). Ergodicity is established, with strong Feller property and $L^p_\mu$ consistency for all $\widetilde{T}$0.

Explicitly, for $\widetilde{T}$1,

$\widetilde{T}$2

ensuring convergence to equilibrium. The Dirichlet form is transient, while the Neumann form is recurrent, aligning with corresponding geometric intuition.

Spectral Theory and Linear Asymptotics

Both Dirichlet and Neumann OU operators have compact resolvents with eigenvalues accumulating only at infinity. Spectral asymptotics are computed using decoupling arguments and index shifting:

• If $\widetilde{T}$3 is the number of infinite edges (half-lines), the $\widetilde{T}$4-th Dirichlet and Neumann eigenvalues admit the asymptotic law:

$\widetilde{T}$5

The leading asymptotics are governed entirely by the structure and number of infinite half-lines, with the compact core yielding only lower-order corrections.

Decomposition and Spectral Analysis on Regular Rooted Trees

A central contribution is the reduction of the spectral problem for the OU operator on regular rooted trees to a family of one-dimensional (radial) half-line OU problems, via an orthogonal decomposition aligned with the radial symmetry.

• Decomposition: $\widetilde{T}$6 is decomposed according to radial and oscillatory branching modes, with the OU operator restricted to invariant subspaces isomorphic to $\widetilde{T}$7 with Gaussian-weighted measure and transmission conditions at generation points.
• Block Reduction: For a regular rooted tree with branching sequences $\widetilde{T}$8 and generation points $\widetilde{T}$9, the Dirichlet operator is unitarily equivalent to an orthogonal sum:

$\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$0

where $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$1 are one-dimensional OU operators on $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$2 with Dirichlet boundary condition, and $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$3 counts the number of branches at generation $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$4.

• Refined Spectral Bounds: The spectrum of the full operator is localised between the lowest and highest "block" eigenvalues, yielding sandwich-type inequalities. For each $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$5,

$\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$6

and analogous bounds for Neumann eigenvalues.

• Spectral Estimates: Asymptotics of the one-dimensional half-line OU eigenvalues are given by

$\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$7

confirming linear growth with explicit error terms depending on radial distance from the root.

Wentzell-Type Realisations and Dynamic Boundary Conditions

The study briefly analyzes operators coupled to boundary dynamics (Wentzell-type conditions) at the root within an augmented Hilbert space. The resulting semigroup is analytic, contractive, and Markovian, providing the solution to a parabolic equation with dynamic transmission at the root and retaining the invariant measure structure.

Discussion and Implications

The rigorous analysis and operator reductions clarify how the infinite geometry of the network dominates the spectral profile of OU generators on metric trees. The branching structure is cleanly encoded in spectral multiplicities. The identified scaling law—linear in $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$8 with inverse proportionality to the number of infinite branches—is robust to the particular choice of vertex conditions within the compact core, as long as self-adjointness is retained.

The results enable sharper understanding of heat kernel behavior, long-term dynamics, and ergodic properties for diffusive processes with unbounded drift on complex/tree-like graphs, extending classical OU theory from $\mu(dx) = (2/\sqrt{\pi}) e^{-x^2} dx$9 to metric tree frameworks. Potential future work includes extension to non-Gaussian weights or variable drift, the study of associated stochastic processes (e.g., exit times, hitting probabilities), and analysis under random or fractal tree geometries.

Conclusion

This paper establishes comprehensive operator-theoretic and spectral results for Ornstein–Uhlenbeck semigroups on rooted metric trees. The explicit reduction to radial problems on regular trees yields precise spectral asymptotics, with computation of strong lower and upper bounds and a clear link to the combinatorial tree structure. The methods and results provide significant groundwork for further analysis of diffusions with unbounded drift on networks, and are likely to facilitate advances in the mathematical theory of stochastic processes and PDEs on graphs.

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Reference: "Ornstein--Uhlenbeck semigroup on rooted trees" (2606.31713)