Lookdown Construction & Sampling Duality

• Lookdown construction and sampling duality is a rigorous framework that represents stochastic population models via ordered particle systems while capturing genealogical ancestry.
• It leverages Poisson processes, level indexing, and explicit Markovian duality to model complex phenomena such as skewed offspring, spatial structure, and selection.
• The duality framework provides actionable insights into fixation probabilities, block-count distributions, and the evolution of measure-valued processes under diverse biological conditions.

The lookdown construction and sampling duality together provide the foundational mathematical framework underpinning modern pathwise and genealogical representations of measure-valued processes such as the Λ-Fleming–Viot, Wright–Fisher, and their generalizations to structured, spatial, or seed-bank models. The lookdown construction realises stochastic population models as infinite or large ordered particle systems whose evolution directly encodes genealogical ancestry, while sampling duality establishes an explicit Markovian link between the forward-in-time evolution of type frequencies and the backward-in-time evolution of genealogical partitions or coalescents. Together, these techniques characterize both the marginal law of types and the distribution of genealogies, extending to regimes involving complex phenomena such as skewed offspring, selection, recombination, dormancy, and environmental conditioning.

1. Formal Structure of the Lookdown Construction

The lookdown construction represents population processes via particle systems indexed by levels, with each level carrying a type label and additional structure as required by the model class (e.g., seed bank status, spatial location). For a finite population of size $N$, individuals are assigned to levels $1,2,\dots,N$, and the evolution is described by Poissonian events:

• Small reproduction events: Each unordered pair $\{i,j\}$ is assigned a Poisson clock, modeling pairwise mergers (e.g., Kingman’s coalescent, with rate $a_0$).
• Large reproduction events: Given by a Poisson random measure over event times, reproduction size $r$, and uniform auxiliary random variables, modeling simultaneous multiple-parent mergers governed by a finite measure $\Lambda$ or its generalizations. In each event, active participants are identified and the lowest-level participant is designated the parent, whose type is immediately copied by other included levels.
• Seed-bank or dormancy structure: Levels may carry a binary activity indicator (active/dormant), with Poissonian switches between states at specified rates $\alpha$, $\sigma$, governing transitions to and from the seed-bank component (Fittipaldi et al., 25 Nov 2025).
• Selection or competitive events: In selective models, additional Poisson processes encode death of disadvantaged type individuals and resampling mechanisms. Selective lookdown spaces, with state-dependent thinning of possible events, characterize the effective rate of attachment to particular genealogies (Blancas et al., 2021, Bah et al., 2013).

Exchangeability and the Lebesgue-almost everywhere control required for well-posedness are established via coupling arguments and thinning of event Poisson processes, allowing the infinite-level limit to be constructed and for the empirical frequency process to converge to a measure-valued diffusion (e.g., Fleming–Viot, Wright-Fisher, $\Lambda$-processes) (Labbé, 2011, Fittipaldi et al., 25 Nov 2025).

2. Pathwise Genealogy and Partition Flows

The lookdown construction induces a pathwise coding of the genealogy by following, for each current level, the time-ordered sequence of reproduction events that imposed a type change. The sequence of events across levels is naturally encoded in a stochastic flow of partitions:

• Each reproduction event corresponds to a partition of the affected levels, which are merged (their ancestry is joined) at that time.
• The sequence of such partitions (and their cocycle property) forms a stochastic flow of partitions, which in law coincides (under appropriate scaling and initial conditions) with the flow of the genealogical Λ-coalescent (Labbé, 2011).
• The pathwise construction ensures that at each time, the ancestral partition of any sample is explicitly recorded, resolving the ancestry not just for the empirical frequencies but for sampled subpopulations or individuals.

Well-posedness in the infinite-level limit depends on existence of asymptotic frequencies and, in processes admitting infinitely many Eves (dominant lineages), on the construction of flows of bridges, as established for Λ-Fleming–Viot processes by Bertoin & Le Gall and further clarified in (Labbé, 2011).

3. Sampling Duality: Moments and Ancestral Processes

Sampling duality provides a functional analytic and probabilistic correspondence between the forward process of type frequencies or measures and the genealogy of sampled individuals:

• Moment duality: For the empirical frequency (type process) $X_t$ and the block-count process of the coalescent $\#\Pi_t$, the duality is given by

$\mathbb{E}_x[X_t^n] = \mathbb{E}_n[x^{\#\Pi_t}]$

where the generator of $X_t$ is dual (in the algebraic sense) to the generator of the ancestral coalescent (Bah et al., 2013, Labbé, 2011, Fittipaldi et al., 25 Nov 2025).

• Functional duality: For sample functions $f$ and initial law $\mu$,

$$\mathbb{E}_\mu[f(\text{sample types at time } t)] = \mathbb{E}_\pi[f(\text{sampled types at time } 0 \text{ associated to }\pi_t)],$$

where $\pi_t$ is the genealogical partition at time 0 induced by the (possibly structured or spatial) Λ-coalescent run backward from time $t$ (Labbé, 2011).

• Structured and spatial extensions: In models with seed banks, spatial structure, or other complexity, the dual process may involve additional structure such as an environmental switching chain ($\xi$), block-marking, or spatial motion and coalescence, leading to structured coalescents and spatial genealogical processes (Fittipaldi et al., 25 Nov 2025, Veber et al., 2012).

Sampling duality serves as the key method for computing fixation probabilities, block-count distributions, and for establishing the probabilistic equivalence between marginal frequency evolution and the sample-ancestry process (Labbé, 2011, Bah et al., 2013, Veber et al., 2012).

4. Conditioning by Change of Measure and Pathwise Particle Representation

Conditioning stochastic population processes—most notably on fixation—presents significant technical challenges in the stochastic analysis of measure-valued diffusions. Traditional approaches via Doob $h$-transforms are analytically tractable but lack an explicit genealogical or particle representation. In contrast, the lookdown construction naturally facilitates explicit conditioning at the pathwise level (Fittipaldi et al., 25 Nov 2025):

• Conditioning on the fixation of a type (e.g., event $F = \{\lim_{t\to\infty} [Z_1(t)+Z_2(t)] =1\}$) is realized by fixing the type at the lowest level (e.g., $X_1(0)=\heartsuit$).
• For the remaining process, every event that would have involved level 1 (the fixed type) instead produces a coordinated mutation event on the lower levels, with the environmental activity indicator $\xi$ controlling the mutation rates and acting as a random environment.
• The empirical frequency dynamics of the remaining levels are governed by a diffusion with a coordinated mutation term, resulting in a pathwise construction for the law of the process conditioned on fixation, with the genealogy corresponding to a Λ-coalescent with block deletions (representing coordinated mutations) controlled by the environmental chain.
• This provides a direct individual-based probabilistic model for the h-transform, preserving the dual genealogical process and furnishing rigorous results for processes with skewed offspring, dormancy (seed-bank), and switching environments (Fittipaldi et al., 25 Nov 2025).

5. Spatial and Selective Extensions

In spatially structured populations, the lookdown construction is implemented by assigning to each particle both a level and a spatial coordinate, with reproduction events localized in space and involving only particles within a given region (Veber et al., 2012):

• Spatial Λ-Fleming–Viot process: Reproduction events are governed by a Poisson point process over space-time-radius-impact, with each event producing local resampling and ancestral merger (coalescence of genealogical lines) in the affected region (Veber et al., 2012).
• Spatial lookdown genealogy: The resulting genealogical process for a finite sample is a spatial coalescent with migration, with ancestries of sampled individuals explicitly constructed via level ordering and local parent selection.

In models with selection and competition, a selective lookdown space is constructed by state-dependent thinning of Poissonian event candidates; genealogical updates at so-called "active" events are implemented via sampling from the selectively thinned space (Blancas et al., 2021). This produces modified coalescent genealogies reflecting the altered probabilities of ancestral lineages under selection pressure.

6. Unified Perspective and Connections

The lookdown construction and sampling duality serve as the unifying framework facilitating the following:

• Explicit pathwise construction of both marginal frequency processes and ancestral genealogies, applicable to a wide variety of measure-valued models (Λ-Fleming–Viot, seed bank, spatial, selective, etc.).
• Rigorous establishment of duality relations, martingale characterizations, and convergence of finite-particle approximations to infinite-population limits.
• Algorithmically tractable representations of genealogies for both forward– and backward–in–time analyses, critical for the paper of fixation, block-count statistics, structured coalescents, and the impact of conditioning (Doob h-transform).
• Transparent encoding of process features such as coordinated block deletions (reflecting conditioning or mutation), spatial migration, environmental stochasticity, and the interplay of selection, competition, and demography (Fittipaldi et al., 25 Nov 2025, Veber et al., 2012, Blancas et al., 2021).

Primary references: (Fittipaldi et al., 25 Nov 2025, Bah et al., 2013, Labbé, 2011, Blancas et al., 2021, Veber et al., 2012).