Feller-Dynkin Boundary Process

• Feller-Dynkin boundary process is an infinite-dimensional Markov process defined as the projective limit of finite-dimensional Feller semigroup processes with coherent cotransition kernels.
• It utilizes intertwiners and coherent families to construct a unique boundary process that encapsulates analytic, probabilistic, and representation-theoretic properties.
• The process exhibits unique invariant measures, often manifesting as determinantal structures, underpinning applications in random matrix theory and potential analysis.

A Feller-Dynkin boundary process is a Markov process on an infinite-dimensional boundary (projective or Martin boundary) constructed as a limit of finite-dimensional Markov processes whose transition operators are Feller semigroups and are consistent with respect to suitable cotransition (interlacing or projection) kernels. Such processes play a central role in infinite-dimensional probability, representation theory, integrable systems, and the analysis of long-term and spectral properties of stochastic models with hierarchical or projective structures.

1. Abstract Setup: Projective Systems and Coherent Families

Let $\{E_N\}$ be a sequence of locally compact (typically countable) Markov state spaces, each equipped with a Feller (strongly continuous, positivity-preserving, contraction) semigroup $T_t^N$ on $C_0(E_N)$, the space of continuous functions vanishing at infinity. Assuming the existence of Markov kernels (links) $\{\Lambda_N^M: E_M \to \mathcal{P}(E_N)\}_{M>N}$ satisfying the projective system properties, a projective family of semigroups is one where these semigroups are compatible: $$T_t^{N+1} \circ \Lambda_N^{N+1} = \Lambda_N^{N+1} \circ T_t^N \qquad \text{for all } t \ge 0.$$ If, in addition, the semigroups and links are Feller (preserve $C_0$ structure), the projective system admits a boundary space $\Omega$, realized as the extreme points of the inverse limit or as an explicit simplex of parameters. The Feller-Dynkin boundary process is the canonical infinite-dimensional Markov process on $\Omega$ whose finite-dimensional projections coincide with those of the coherent family—i.e., for all $N$,

$$T_t^{\Omega} \circ \Lambda_N^\Omega = \Lambda_N^\Omega \circ T_t^N, \qquad t \geq 0.$$

Here $\Lambda_N^\Omega$ denotes the limit of the links $\Lambda_N^M$ as $M \to \infty$ mapping $\Omega$ to $E_N$.

The general existence and uniqueness of the boundary process are established using standard arguments from semigroup theory and the theory of Feller boundaries (Borodin et al., 2010, Bufetov et al., 21 Sep 2025).

2. Canonical Examples: Gelfand–Tsetlin and Weyl Chamber Hierarchies

2.1 Gelfand–Tsetlin Graph and Unitary Characters

The Markov processes constructed in (Borodin et al., 2010) are defined on the path space of the Gelfand-Tsetlin graph, which encodes the representation-theoretic branching rules for $U(N)$ as $N$ increases. At each finite $N$, the state space $GT_N$ consists of signatures $\lambda = (\lambda_1 \geq \cdots \geq \lambda_N)$, with transition rates given by explicit Doob $h$-transforms (Vandermonde weights) of independent birth–death processes: $$D^{(N)}(X,V) = \frac{\operatorname{Dim}_N(V)}{\operatorname{Dim}_N(X)} \left\{ \sum_{i=1}^N D(x_i, v_i) 1_{\{\lambda_i(V) - \lambda_i(X) = \pm 1\}} - d_N \right\}$$ where $X, V \in GT_N$ and $D(x, y)$ is the rate matrix for the one-dimensional process, parameterized by a quadruple $(z, z', w, w')$ as in Eq. (5.1) of (Borodin et al., 2010).

The processes at each level are linked by the cotransition kernels $A_{N+1}$ corresponding to the branching law $U(N+1) \downarrow U(N)$. The consistent system of semigroups thus defines a unique Feller process on the boundary $\Omega$ of the Gelfand-Tsetlin graph, realized concretely as the space of extreme characters of $U(\infty)$.

2.2 Infinite Weyl Chambers and Interlacing Diffusions

The projective system also appears in the context of Weyl chambers $$W_N = \{ x \in \mathbb{R}^N: x_1 \leq \cdots \leq x_N \}$$ with links given by partial symmetrization or eigenvalue projection, as in the construction for Hua–Pickrell diffusions (Assiotis, 2017) or Laguerre/Pickrell processes (Bufetov et al., 21 Sep 2025). In those examples, one considers diffusions or SDEs with explicit drift and diffusion coefficients, such as: $$dX_t^{N,i} = \sqrt{2 X_t^{N,i}}\, dB_t^i + [-X_t^{N,i} + \alpha + N + \sum_{j\neq i} \frac{X_t^{N,i} + X_t^{N,j}}{X_t^{N,i} - X_t^{N,j}}] dt$$ for the Laguerre family.

3. Construction via Intertwining and the Method of Intertwiners

The method of intertwiners is central: it consists in establishing explicit intertwining relations between semigroups on different levels and Markov kernels connecting state spaces. Given Feller semigroups $T_t^N$, links $\Lambda_N^{N+1}$, and the property: $$T_t^{N+1} \circ \Lambda_N^{N+1} = \Lambda_N^{N+1} \circ T_t^N$$ one can (under suitable analytic assumptions) apply abstract projective limit theorems (see e.g. Proposition 2.4 of (Borodin et al., 2010) and Definition 2.2 of (Bufetov et al., 21 Sep 2025)) to construct a unique Markov semigroup $T_t^\Omega$ acting on $C_0(\Omega)$ that projects properly to all finite $N$.

In the classic cases, the system's combinatorial or representation-theoretic structure provides:

• explicit kernels $\Lambda$
• determinantal or totally positive transition matrices
• contraction and Feller properties

that ensure the projective limit exists and is unique.

4. Properties: Invariant Measures and Determinantal Structure

For these boundary processes, one central property is the existence of a unique invariant measure, often arising as the projective limit of finite-level invariant (Gibbs or central) measures. In the Gelfand–Tsetlin context, the so-called $zw$-measure

$$M_{z,z',w,w'}^{(N)}(X) \propto \prod_{1 \leq i < j \leq N} (\lambda_i - \lambda_j + j - i)^2 \prod_{i} W_{z,z',w,w'}(\lambda_i + N - i)$$

is consistent and projects to a unique $M_{z,z',w,w'}$ on $\Omega$. This measure is also interpretable as the spectral measure in harmonic analysis for $U(\infty)$.

Borel summation and pushforward maps show that this invariant measure is a determinantal point process with explicit correlation kernel given in terms of the Gauss hypergeometric function; see [(Borodin et al., 2010), Sec. 8] and related works on determinantal ensembles.

In some boundary processes, such as for Laguerre processes, the infinite-dimensional limit (“boundary process”) can become deterministic: $$\alpha_i(t) = \alpha_i(0) e^{-t},\qquad \gamma(t) = 1 + (\gamma(0) - 1) e^{-t}$$ as shown in Theorem 1.4 of (Bufetov et al., 21 Sep 2025).

For Pickrell and related diffusions, the infinite-dimensional boundary retains stochasticity, and the invariant measure is identified with the ergodic decomposition of (generalized) Pickrell measures.

5. Connections with Martin Boundaries and Potential Theory

Feller-Dynkin processes realize the projective (or Martin) boundary in a probabilistic framework. The process on the boundary (i.e., the Feller-Dynkin process) encodes the “extreme” harmonious behavior associated with the minimal or extremal harmonic functions in potential theory. In many cases, the boundary process arises as the unique Feller extension of a Markov process to a completed state space, and unique invariant measures are characterized via harmonic analysis (cf. (Borodin et al., 2010, Kim et al., 2015)).

When the boundary is accessible in the sense of potential theory (for instance, if

$\int_D G_D(x, y) m(dy) = \infty$

for all $x$), then the Martin boundary at infinity contains a unique minimal point, and the boundary process can be realized explicitly as the limit of finite processes with increasingly “spread out” initial data (Kim et al., 2015).

6. Representation-Theoretic and Integrable Structures

These constructions are tightly linked to deep structures in representation theory (characters of infinite-dimensional groups), integrable systems (determinantal processes, orthogonal polynomial ensembles), and random matrix theory:

• The Markov dynamics respect central or Gibbs measures (arising from representation-theoretic harmonic analysis).
• The invariant (spectral) measures correspond to determinantal point processes with explicit correlation kernels (e.g., Christoffel–Darboux/Hermite/hypergeometric).
• Bulk and edge scaling limits of the boundary processes yield universal objects (sine kernel, Airy, etc.).

Moreover, in the infinite-dimensional limit, some processes “wash out” randomness, yielding deterministic flows (as in infinite Laguerre limits), whereas others preserve nontrivial ergodic components (as in Pickrell or Hua-Pickrell diffusions).

7. Analytical and Probabilistic Implications

The Feller-Dynkin boundary process formalism unifies several perspectives:

• Analytical: It provides a semigroup-theoretic and generator-based framework that links spectral properties, existence/uniqueness results, and boundary conditions for infinite-dimensional processes.
• Probabilistic: It enables a pathwise and measure-theoretic construction for infinite particle systems and processes arising from interlacing, projective, or branching structures.
• Applications: It leads to explicit constructions and invariant measures for models in random matrix theory, infinite combinatorial settings, and stochastic representations of harmonic functions and characters.

The approach also clarifies the role of intertwiners and boundary behavior in the scaling and limiting behavior of integrable Markovian systems.

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This synthesis delineates the conceptual framework, technical construction, explicit dynamical properties, invariance, and deep connections to both potential theory and representation/integrable systems that define Feller-Dynkin boundary processes as studied in (Borodin et al., 2010, Bufetov et al., 21 Sep 2025, Assiotis, 2017), and related works.