Fleming–Viot Diffusion Overview

• Fleming–Viot diffusion is a measure-valued stochastic process that models allele frequency evolution in infinite populations via resampling and mutation.
• It connects classical Wright–Fisher dynamics with interacting particle system approaches for approximating quasi-stationary distributions in conditioned processes.
• Recent extensions apply its framework to Bayesian nonparametrics and diffusion-model alignment, enhancing predictive inference and efficient resampling in machine learning.

Fleming–Viot diffusion (FVD) denotes a class of probability-measure-valued stochastic processes whose core mechanisms are resampling and, depending on the model, mutation, killing, rebirth, or immigration. In the classical population-genetics sense, it is a measure-valued Markov process on a space of probability measures and is closely related to Wright–Fisher diffusion. In another major line of work, “Fleming–Viot” refers to finite-$N$ particle systems with killing and instantaneous rebirth, used to represent conditioned processes and quasi-stationary distributions; their hydrodynamic or scaling limits can themselves produce law-valued processes of Fleming–Viot type (Tough, 2021, Tough et al., 2020). The literature therefore treats FVD both as a foundational measure-valued diffusion and as the limit object or conceptual analogue of interacting particle approximations.

1. Terminology, state spaces, and competing usages

In the population-genetics tradition, a Fleming–Viot diffusion is a $\mathcal{P}(\mathbb{K})$- or $\mathcal{M}_1(E)$-valued Markov process, where $\mathbb{K}$ or $E$ is a Polish type space. It describes the evolution of allele-frequency distributions in an idealized infinite population under random resampling and, in many formulations, mutation. In the simplest neutral setting, the state at time $t$ is a probability measure $Z_t$ or $\nu_t$, and the process has càdlàg or continuous paths depending on the mutation and resampling mechanism (Hughes et al., 2022, Tough, 2021).

A distinct but closely connected usage arises in the theory of conditioned processes. There, a Fleming–Viot particle system consists of $N$ particles evolving in a domain $U$ or $\mathcal{P}(\mathbb{K})$0, with the rule that when a particle hits the killing boundary, it is instantaneously reborn at the position of another particle chosen uniformly among the survivors. The empirical measure

$\mathcal{P}(\mathbb{K})$1

then approximates the law of the underlying killed process conditioned on survival, and stationary empirical measures approximate quasi-stationary distributions (QSDs) (Tough et al., 2020, Journel et al., 2022).

The literature explicitly distinguishes these senses. One formulation identifies “Fleming–Viot measure-valued diffusions (population genetics)” and “Fleming–Viot particle systems for conditioned processes” as two related but distinct notions. A further, contemporary reuse appears in machine learning, where “Fleming-Viot Diffusion (FVD)” is the name of an inference-time alignment method for diffusion models based on an FV-style birth–death resampling mechanism; this is inspired by, but not identical to, the classical stochastic-process object (Tough et al., 2020, Shekhar et al., 8 Apr 2026).

2. Classical measure-valued diffusion and the Wright–Fisher connection

A standard rigorous definition of the classical Fleming–Viot diffusion is via a martingale problem on polynomial functionals of the form

$\mathcal{P}(\mathbb{K})$2

with generator

$\mathcal{P}(\mathbb{K})$3

Here $\mathcal{P}(\mathbb{K})$4 is the resampling rate, $\mathcal{P}(\mathbb{K})$5 merges coordinates $\mathcal{P}(\mathbb{K})$6 and $\mathcal{P}(\mathbb{K})$7, and the process $\mathcal{P}(\mathbb{K})$8 takes values in $\mathcal{P}(\mathbb{K})$9. The corresponding martingale problem is well-posed, and paths are continuous in the weak topology and in the weak-atomic metric (Tough, 2021).

Finite-dimensional projections recover the classical Wright–Fisher diffusion. If $\mathcal{M}_1(E)$0 is a measurable partition and $\mathcal{M}_1(E)$1, then $\mathcal{M}_1(E)$2 is an $\mathcal{M}_1(E)$3-type Wright–Fisher diffusion with generator

$\mathcal{M}_1(E)$4

This relation makes the measure-valued Fleming–Viot process the infinite-type extension of Wright–Fisher dynamics (Tough, 2021).

A complementary representation rewrites the $\mathcal{M}_1(E)$5-Fleming–Viot generator in Wright–Fisher form evaluated at a random affine transform of the current frequency vector. In the two-allele case,

$\mathcal{M}_1(E)$6

which has the same quadratic form $\mathcal{M}_1(E)$7 as the classical Wright–Fisher generator but with derivatives evaluated at a random transformed state. In the degenerate case $\mathcal{M}_1(E)$8, the representation reduces to the ordinary Wright–Fisher generator (Griffiths, 2012). This formulation makes the spectral structure transparent: the eigenvalues encode coalescent merger rates, and the eigenfunctions become $\mathcal{M}_1(E)$9-deformations of Jacobi-type polynomials.

3. Generalizations: $\mathbb{K}$0-resampling, immigration, and support propagation

The $\mathbb{K}$1-Fleming–Viot process generalizes binary resampling to multiple-merger reproduction events. Let $\mathbb{K}$2, and define coalescent merger rates

$\mathbb{K}$3

In the neutral setting with mutation operator $\mathbb{K}$4, the $\mathbb{K}$5-Fleming–Viot process is a $\mathbb{K}$6-valued Markov process with generator built from a mutation part $\mathbb{K}$7 and a resampling part indexed by subsets $\mathbb{K}$8 of size at least $\mathbb{K}$9 (Hughes et al., 2022). Backward in time, its genealogy is the $E$0-coalescent rather than the Kingman coalescent.

For Lévy mutation, the process exhibits an instantaneous support phenomenon. If $E$1 is a $E$2-Fleming–Viot process with Lévy mutation measure $E$3, then for every $E$4,

$E$5

almost surely, and if $E$6, then $E$7 almost surely for all $E$8 (Hughes et al., 2022). The proof uses the Donnelly–Kurtz lookdown representation and decomposes the population into ancestral clusters.

Immigration leads to another extension. In the framework of $E$9-generalized Fleming–Viot processes with immigration, the state remains a probability measure, but new mass of distinguished immigrant type $t$0 is introduced by replacement events. For suitable stable continuous-state branching processes with immigration, a time-changed ratio process becomes an $t$1-generalized Fleming–Viot process whose parameters are Beta laws. In particular, in the stable case one obtains

$t$2

and the genealogy is described by an $t$3-coalescent; in the conditioned-to-never-be-extinct case, this becomes a $t$4-coalescent (Foucart et al., 2012).

These generalizations show that the classical binary-resampling diffusion is only one point in a larger family of measure-valued processes parameterized by coalescent structure, mutation mechanism, and immigration dynamics. A plausible implication is that many qualitative path properties of classical FVD survive outside the Kingman/Brownian regime, but with genealogical and spectral data controlled by the relevant $t$5- or $t$6-measure.

4. Fleming–Viot particle systems, conditioned laws, and quasi-stationarity

The interacting-particle formulation begins from a killed Markov process. In a domain $t$7, one lets $t$8 particles diffuse independently until a particle hits $t$9, at which time it is resurrected at the position of another particle chosen uniformly among the survivors. The empirical measure remains a probability measure on $Z_t$0, and as $Z_t$1 it approximates the conditional law of the killed process given survival (Tough et al., 2020).

For a killed McKean–Vlasov diffusion,

$Z_t$2

the conditioned law

$Z_t$3

is the hydrodynamic limit of the corresponding Fleming–Viot particle system with empirical drift dependence, and the cumulative killing rate

$Z_t$4

is approximated by the normalized jump count

$Z_t$5

The hydrodynamic limit theorem states that

$Z_t$6

in the topology of uniform convergence on compacts in time, in probability (Tough et al., 2020).

A QSD for the killed process is a probability measure $Z_t$7 such that

$Z_t$8

where $Z_t$9. In the McKean–Vlasov setting, if $\nu_t$0 is a QSD, then there exists $\nu_t$1 such that

$\nu_t$2

and if $\nu_t$3, then $\nu_t$4 (Tough et al., 2020). The same paper also gives an explicit non-uniqueness example on $\nu_t$5 with mean-field drift, where the QSD set contains three distinct symmetric/asymmetric densities corresponding to a pitchfork bifurcation.

Selection among multiple QSDs is a recurring theme. For a subcritical continuous-time Galton–Watson process with absorption at $\nu_t$6, the associated discrete Fleming–Viot particle system has a unique invariant measure $\nu_t$7, and its stationary empirical distribution converges, as $\nu_t$8, to the minimal quasi-stationary distribution $\nu_t$9 of the Galton–Watson process (Asselah et al., 2012). For Brownian motion with drift $N$0 on $N$1, killed at $N$2, the killed process has an infinite family of QSDs $N$3, but the empirical measure of the $N$4-particle stationary distribution converges to the minimal QSD $N$5, and the stationary jump rate satisfies $N$6 (Tough, 2023).

Quantitative convergence results are also available in discrete space. For a discrete Fleming–Viot or Moran-type system on a countable state space $N$7 with absorbing cemetery state $N$8, if

$N$9

then the $U$0-particle system contracts exponentially in a Wasserstein distance induced by

$U$1

and the conditioned semigroup converges exponentially fast to a unique QSD. Two-particle correlations are $U$2, and the empirical process converges uniformly in time to the conditioned process with an explicit rate (Cloez et al., 2013).

Hard killing can produce both stability and failure modes. In a metastable diffusion killed on $U$3, the long-time convergence of the Fleming–Viot system toward a stationary measure occurs at an exponential rate independent of $U$4, together with uniform-in-time propagation of chaos estimates (Journel et al., 2022). By contrast, for certain one-dimensional diffusions with strong inward drift, the system can go extinct in finite time: with two particles driven by a Bessel process on $U$5, both particles converge to $U$6 at a finite time if and only if the Bessel dimension is less than $U$7, and for Brownian motion with sufficiently strong inward drift on $U$8, extinction occurs for any number of particles (Bieniek et al., 2011).

5. McKean–Vlasov, control, and scaling links between the two traditions

The conditioned-process and population-genetics traditions are not disjoint. One bridge is the Fleming–Viot multi-colour process. Consider an $U$9-particle Fleming–Viot system associated to a normally reflected diffusion with soft catalyst killing, and attach colours $\mathcal{P}(\mathbb{K})$00 to the particles. At each killing event, the dead particle inherits both the position and colour of a uniformly chosen survivor. Writing

$\mathcal{P}(\mathbb{K})$01

the paper proves that after rescaling time by $\mathcal{P}(\mathbb{K})$02, the colour empirical process converges to the population-genetics Fleming–Viot diffusion. More precisely, if the tilted empirical initial measures converge to $\mathcal{P}(\mathbb{K})$03, then

$\mathcal{P}(\mathbb{K})$04

where $\mathcal{P}(\mathbb{K})$05 is a Wright–Fisher/Fleming–Viot process of rate

$\mathcal{P}(\mathbb{K})$06

Here $\mathcal{P}(\mathbb{K})$07 is the principal eigentriple of the killed spatial generator (Tough, 2021). This result shows that the genetic information carried by a conditioned-process Fleming–Viot system has a scaling limit given by the classical measure-valued Fleming–Viot diffusion.

A second bridge is control. In a bounded domain $\mathcal{P}(\mathbb{K})$08, a controlled diffusion can be killed on exit and evaluated through the survival-conditioned law

$\mathcal{P}(\mathbb{K})$09

Under boundedness, Lipschitz continuity in total variation, invertible $\mathcal{P}(\mathbb{K})$10, and a convexity assumption, open-loop and closed-loop formulations are equivalent. The closed-loop problem is linked to a McKean–Vlasov Fleming–Viot-type dynamics in which, when a particle leaves $\mathcal{P}(\mathbb{K})$11, it is instantaneously reinserted into the domain according to its own current law. The corresponding SDE with jump term is

$\mathcal{P}(\mathbb{K})$12

with the reinsertion law satisfying

$\mathcal{P}(\mathbb{K})$13

The paper proves that this Fleming–Viot McKean–Vlasov SDE has a unique in law weak solution and that

$\mathcal{P}(\mathbb{K})$14

where $\mathcal{P}(\mathbb{K})$15 solves the conditional McKean–Vlasov SDE (Jettkant, 2024).

These results suggest that the phrase “Fleming–Viot diffusion” may designate not only a specific classical generator but also a limit principle: finite-$\mathcal{P}(\mathbb{K})$16 kill-and-clone systems, once enriched by colours, conditioning, or mean-field drift, can converge to measure-valued dynamics of Wright–Fisher or McKean–Vlasov type.

6. Bayesian nonparametrics, predictive inference, and contemporary reinterpretations

Fleming–Viot diffusions also appear as time-dependence mechanisms in Bayesian nonparametrics. On a Polish sampling space $\mathcal{P}(\mathbb{K})$17, let $\mathcal{P}(\mathbb{K})$18 with $\mathcal{P}(\mathbb{K})$19 nonatomic, and let $\mathcal{P}(\mathbb{K})$20 be the Dirichlet process law. A Fleming–Viot-driven dependent Dirichlet process is a Markov process $\mathcal{P}(\mathbb{K})$21 with $\mathcal{P}(\mathbb{K})$22 and transition kernel

$\mathcal{P}(\mathbb{K})$23

where $\mathcal{P}(\mathbb{K})$24 are the transition probabilities of a pure-death process with rates

$\mathcal{P}(\mathbb{K})$25

All time marginals are Dirichlet processes with parameter $\mathcal{P}(\mathbb{K})$26, and the model forms a hidden Markov model with emissions $\mathcal{P}(\mathbb{K})$27 (Ascolani et al., 2020).

This FV-driven construction yields explicit predictive distributions. If $\mathcal{P}(\mathbb{K})$28 denotes past observations and $\mathcal{P}(\mathbb{K})$29 is the finite set of active multiplicity indices from the posterior mixture representation, then the predictive law of the first future observation at time $\mathcal{P}(\mathbb{K})$30 is

$\mathcal{P}(\mathbb{K})$31

The induced partition structure admits a Chinese restaurant process metaphor “with a conveyor belt,” in which new tables may take dishes either from the baseline distribution or from a time-varying subset of dishes inherited from earlier times (Ascolani et al., 2020).

A much more recent reuse of the acronym appears in diffusion-model alignment. “Fleming-Viot Diffusion (FVD)” is introduced as an inference-time alignment method that replaces multinomial SMC resampling with an FV-style birth–death mechanism. Particles survive independently with probabilities

$\mathcal{P}(\mathbb{K})$32

dead particles are reborn by copying uniformly chosen survivors, and extra rebirth noise is added to avoid deterministic trajectory collapse. The method is fully parallelizable and, empirically, “outperforms prior methods by 7% in ImageReward,” “improves FID by roughly 14-20% over strong baselines,” and is “up to 66 times faster than value-based approaches” (Shekhar et al., 8 Apr 2026). This usage is algorithmic rather than measure-valued in the classical probabilistic sense, but its birth–death logic is explicitly modeled on Fleming–Viot population dynamics.

Across these domains, a common structural motif persists: a fixed-size population of weighted or unweighted entities evolves through propagation and resampling, and the resulting empirical distribution is the central object. In classical probability this produces measure-valued diffusions and conditioned-law approximations; in Bayesian nonparametrics it yields dependent random probability measures with tractable predictive structure; and in contemporary diffusion-model alignment it motivates diversity-preserving resampling schemes.