Tree-Valued Diffusions

Updated 3 April 2026

Tree-valued diffusions are Feller–Markov processes that evolve metric-measure trees, explicitly encoding entire genealogical structures in a continuous state space.
They employ state spaces like measure-metric and ultrametric trees along with topological frameworks such as the Gromov–Prohorov and Gromov–Hausdorff–Prokhorov metrics to ensure continuity and measurability.
Methodologies involving martingale problems, dualities, and interval-partition diffusions underpin their construction, linking discrete models with continuum random trees and ensuring robust genealogical modeling.

A tree-valued diffusion is a Feller–Markov process taking values in an isometry or measure-preserving isomorphism class of random metric-measure trees and modeling the evolving genealogy of populations under reproduction, selection, mutation, and other evolutionary forces. These processes generalize classical measure-valued diffusions by explicitly encoding the full (potentially infinite) tree-structured genealogy in a continuous state space, typically equipped with variants of the Gromov–Prohorov or Gromov–Hausdorff–Prokhorov topology. Tree-valued diffusions arise as scaling limits of discrete random tree chains and are intimately connected to interval-partition diffusions, exchangeable coalescent processes, continuum random trees (CRTs), and measure-valued Feller or Fleming–Viot diffusions.

1. State Spaces and Topological Frameworks

The foundational aspect of tree-valued diffusions is the specification of an appropriate state space and topology that ensure tractability, continuity, and measurability.

Measure-metric trees: The space 𝕄 of isomorphism classes of (complete, separable) metric-measure trees with the Gromov–Prohorov metric $d_p$ , and its compact version 𝕄_c with the Gromov–Hausdorff–Prokhorov metric $d_{GHP}$ . Here, a state is $[X, r, \mu]$ where $(X, r)$ is a metric space and $\mu$ a probability measure.
Ultrametric measure spaces: The space $\mathbb{U}$ of equivalence classes $[U, r, \mu]$ for $(U, r)$ ultrametric and $\mu$ finite Borel, modulo measure-preserving isometries. The Gromov–Prohorov topology generates a Polish space (Depperschmidt et al., 2019).
Marked ultrametric spaces: For multitype models, marked versions $\mathbb{U}^I$ employ an additional mark space $d_{GHP}$ 0, tracked using the marked Gromov-weak topology (Depperschmidt et al., 2011).
Interval partition spaces: Certain tree diffusions use interval-partition representations, encoding the mass distribution of branching subtrees, endowed with diversity metrics $d_{GHP}$ 1 (e.g., infimum of block and diversity discrepancies) (Forman et al., 2016).

These structures support continuous-path Feller processes and accommodate immersed jump structures or “dust” via marked spaces as needed in non-compact genealogies (Gufler, 2014).

2. Martingale Problems, Generators, and Duality

Tree-valued diffusions are rigorously characterized by martingale problems for generators decomposed into growth, resampling (coalescence), mutation, and selection terms.

Polynomial test functions: Most constructions use polynomials in sampled distances (“distance-matrix polynomials”), e.g.

$d_{GHP}$ 2

where $d_{GHP}$ 3 maps points to their mutual distance matrices (Depperschmidt et al., 2019, Depperschmidt et al., 2011).

Canonical generator: For tree-valued Feller diffusion, the generator is

$d_{GHP}$ 4

where

$d_{GHP}$ 5

and the branching part merges sampled leaves at a rate proportional to coalescence (Depperschmidt et al., 2019).

Enriched models: Fleming–Viot processes with mutation and selection further enrich the generator by mutation and selection operators acting on marks (Depperschmidt et al., 2011).
Martingale property: For any test polynomial $d_{GHP}$ 6, the process

$d_{GHP}$ 7

is a martingale, defining the process unambiguously.

Dualities (e.g., to distance-matrix-augmented coalescents or function-valued processes) are used to prove uniqueness and ergodicity; see (Depperschmidt et al., 2019, Depperschmidt et al., 2011).

3. Interval-Partition Diffusions and Connection to Tree-Valued Processes

Interval-partition-valued diffusions serve as key building blocks or projections for tree-valued processes, particularly in the context of Aldous diffusion and related CRT dynamics:

Scaffolding and spindles: Tree branches or subtrees correspond, via skewer maps, to evolving interval partitions whose block sizes encode subtree masses. These partitions themselves evolve as Feller–Markov processes driven by Lévy (e.g., Stable( $d_{GHP}$ 8)) or Crump-Mode-Jagers processes decorated with BESQ excursions (Forman et al., 2016, Forman et al., 2019).
Poisson–Dirichlet laws: Stationary distributions on partitions are Poisson–Dirichlet $d_{GHP}$ 9 or $[X, r, \mu]$ 0, corresponding to special cases of Petrov’s EKP diffusions for ranked masses (Forman et al., 2016).
Pathwise tree projections: The sequence of interval-partition evolutions attached to the spines and edges of finite (or $[X, r, \mu]$ 1-leaf) trees yields a pathwise construction of the evolving metric-measure structure of CRTs, culminating in the stationary Aldous diffusion (Forman et al., 2023, Forman et al., 2018).
Ray–Knight correspondences: Diversity of interval partitions relates directly to local times of the underlying Lévy scaffolding, establishing a deep connection between interval partition diversity, subtree masses, and genealogical distances (Forman et al., 2019).

4. The Aldous Diffusion and Continuum Random Tree Dynamics

The Aldous diffusion is the Markov process on the Gromov–Hausdorff–Prokhorov space of compact, rooted, weighted real trees conjectured to have the Brownian CRT as unique stationary law (Pal, 2011, Forman et al., 2023, Forman et al., 2018):

Construction: Built via consistent projective systems of stationary $[X, r, \mu]$ 2-tree evolutions, each edge decorated with interval-partition diffusions (types 0, 1, 2 corresponding to branching structures) (Forman et al., 2023, Forman et al., 2018).
Marginals: Mass splits at branch points evolve as multi-dimensional Wright–Fisher diffusions with negative mutation rates (e.g., projection to $[X, r, \mu]$ 3-dimensional mass-simplices governed by killed WF diffusions) (Forman et al., 2018).
Path properties: Exhibits right-continuity and path-continuity in the Gromov–Hausdorff–Prokhorov topology except at a null set of exceptional times with instantaneous ternary coalescence. The process is not a Hunt process due to quasi-left-continuity failure at these times (Forman et al., 2023).
Ergodicity and stationarity: The Aldous diffusion is stationary and ergodic with respect to the law of the Brownian CRT (Forman et al., 2023, Forman et al., 2018).

5. Genealogy-Valued Feller Diffusion and Cox Cluster Representations

The genealogy-valued Feller diffusion models genealogical evolution as a process in the space $[X, r, \mu]$ 4 of ultrametric measure spaces:

Concatenation property: The process satisfies a generalized branching property; subtrees below a certain height evolve independently and can be represented as a Cox point process of sub-family genealogies (Depperschmidt et al., 2019).
Feynman–Kac duality: Uniqueness and existence rely on duality with Kingman coalescent processes augmented with distance dynamics.
Conditioning and long-time limits: Conditioning on survival leads to $[X, r, \mu]$ 5-processes and quasi-equilibrium laws, with rescaling producing limiting distributions for genealogical structures (Kolmogorov–Yaglom limits, Palm decompositions) (Depperschmidt et al., 2019).

6. Fleming–Viot Tree-Valued Processes, Scaling Limits, and Path Properties

Scaling limits of discrete genealogical chains (e.g., Cannings or Moran models) yield tree-valued Fleming–Viot diffusions:

Scaling regime: Under classical Möhle–Sagitov conditions, genealogical chains converge (in the Gromov–Prohorov or marked Gromov–Prohorov topology, depending on dust presence) to Fleming–Viot diffusions in spaces of (possibly marked) ultrametric measure spaces (Gufler, 2016, Gufler, 2014).
Path properties: These processes are càdlàg, with jumps corresponding to large reproduction or extinction events; in the dust-free case, paths are continuous away from these times (Gufler, 2014).
Generators: The growth and resampling parts of the generator act on polynomials in distances, with rates determined by coalescent kernels for the underlying reproduction mechanisms.

7. Ray–Knight Representations and Generalizations

Ray–Knight theorems provide representations of genealogical processes in terms of local times of reflected (or drifted) Brownian paths or more general height processes:

Critical branching: Feller's branching diffusion can be represented as the local time profile of a reflected Brownian motion, each excursion coding a continuum tree, and local time at level $[X, r, \mu]$ 6 corresponding to population mass at time $[X, r, \mu]$ 7 (Le et al., 2013).
Ray–Knight for logistic growth: Extensions to processes with affine local-time-dependent drift yield tree-valued Feller diffusions with logistic growth mechanisms, accommodating selection or competition (Le et al., 2013).

This synthesis presents the key construction principles, metric state spaces, martingale and duality formulations, and recent developments (particularly in pathwise CRT/Aldous diffusion) underpinning contemporary research in tree-valued diffusions (Forman et al., 2023, Forman et al., 2016, Depperschmidt et al., 2019, Forman et al., 2018, Gufler, 2014, Depperschmidt et al., 2011, Le et al., 2013). These advances facilitate a unified treatment of evolving genealogical structures, robustly connecting discrete and continuum population models within a probabilistic, measure-theoretic framework.