CDFI in Stochastic Processes

• The coming-down-from-infinity (CDFI) property defines how stochastic processes starting from an infinite state transition to a finite state in any positive time.
• Researchers establish CDFI using explicit integral and series criteria—such as Schweinsberg’s criterion for Λ-coalescents and summability conditions for birth–death processes—to distinguish processes that remain infinite from those that do not.
• CDFI informs scaling laws and phase transitions by linking deterministic speed functions to genealogical and metric properties, thereby guiding the analysis of coalescent, fragmentation, and competition models.

The coming-down-from-infinity (CDFI) property describes the phenomenon whereby a stochastic population or partition-valued process initialized from an infinite state attains a finite state in any strictly positive time interval. This phenomenon serves as a qualitative demarcation between processes that remain “stuck at infinity” versus those for which infinity is merely an entrance boundary. CDFI criteria have been studied for coalescent processes (particularly Kingman’s coalescent and its generalizations), exchangeable fragmentation–coalescence processes, pure death and birth–death chains, interacting particle systems, diffusion and branching models, and stochastic partial differential equations.

1. Formal Definitions and Foundational Models

Let $(X_t)_{t\geq0}$ be a Markov process on a state space augmented by an infinite state (often $\mathbb{N}_0 \cup \{\infty\}$, partitions of $\mathbb{N}$, or function-valued spaces), with $X_0=\infty$. We say the process comes down from infinity if

$\Pr(X_t < \infty \text{ for all } t > 0) = 1.$

In coalescent and fragmentation–coagulation models, $X_t$ often represents the number of blocks at time $t$. For branching, birth–death, or particle systems, $X_t$ may denote a (possibly infinite) population size or measure.

A representative example is Kingman's coalescent, where any pair of blocks merges at rate $c>0$. Starting from infinitely many singletons, for any $t>0$,

$$\Pr(\text{number of blocks at time } t < \infty) = 1.$$

This property is classical for Kingman’s coalescent and its $\Lambda$-coalescent generalizations with suitable merger rates (Kyprianou et al., 2016, Biehler et al., 2011).

2. Necessary and Sufficient Criteria for CDFI

CDFI is governed by explicit integral or series criteria balancing drift, branching, coalescence, competition, and fragmentation rates:

• Pure death/birth-death chains: The process comes down from infinity iff

$$S = \sum_{n=1}^\infty \pi_n < \infty, \quad \pi_1 = 1, \quad \pi_n = \frac{\lambda_1 \cdots \lambda_{n-1}}{\mu_2 \cdots \mu_n}$$

where $\lambda_n$ are birth rates, $\mu_n$ are death rates (Bansaye et al., 2013, Bansaye et al., 2015). For pure death rates $\lambda_n=0$, the criterion is $\sum_{n=2}^\infty 1/\mu_n < \infty$.

• $\Lambda$-coalescents: Schweinsberg’s criterion: CDFI occurs iff

$$\sum_{b=2}^\infty \frac{1}{\gamma_b} < \infty, \quad \gamma_b = \sum_{k=2}^b k\binom{b}{k} \lambda_{b,k}$$

or equivalently, $\int^\infty dq/\psi(q) < \infty$ for the associated branching mechanism $\psi$ (Biehler et al., 2011, Miller et al., 2016, Liu et al., 8 Jun 2025).

• Fragmentation-coalescence: For EFC processes, there is a phase transition characterized by

$$\theta_\star = \liminf_{n\to\infty} \sum_{k=1}^\infty \frac{n\,\overline\mu(k)}{\Phi(n+k)}, \quad \Phi(n) = \text{coalescent speed function}$$

and CDFI occurs if $\theta^\star < 1$ (Foucart, 2016, Kyprianou et al., 2016).

• CB processes with competition/drift: Integral conditions involve the competition/growth function $g(\cdot)$ or generalized drifts $I(\cdot)$,

$\int_{z_0}^\infty \frac{dy}{g(y)} < \infty$

for competition (Leman et al., 2018), or analogous Lyapunov-based conditions on the drift/branching interaction (Rebotier, 7 Oct 2025).

• Time-changed Lévy and diffusion processes: The condition

$\int^\infty \frac{dx}{x \Psi(1/x) R(x)} < \infty$

where $\Psi$ is the log-Laplace exponent of the input Lévy process and $R(x)$ the time change, ensures CDFI (Foucart et al., 2019, Bansaye et al., 2017).

3. Quantitative Rates and Scaling Laws

When CDFI occurs, the process typically follows a deterministic “speed” function $v(t)$ for small $t$:

• Coalescents: For Kingman, $v(t) = 2/c t$; for Beta$(a,b)$-coalescents ($a<1$), $v(t) \propto t^{1/(a-1)}$.
• Birth–death processes: If death rates are regularly varying, $v(t) \sim ((p-1)t)^{1/(1-p)}$ for $\mu_n \sim n^p$ (Bansaye et al., 2015, Bansaye et al., 2013, Sagitov et al., 2016).
• Fragmentation-coalescence: In the "fast" regime of $c > 2\lambda$, the scaling is $t N(t) \to 2/c$ (Kyprianou et al., 2016).
• Nested coalescents: For nested Kingman, $N(t) \sim 2\gamma/c t^2$ with $\gamma \approx 3.45$, characterizing the joint action of species- and individual-level mergers (Benítez et al., 2018).
• CB with competition: For quadratic competition, solution of $\frac{dz}{dt} = -g(z)$ yields the CDFI speed (Leman et al., 2018, Rebotier, 7 Oct 2025).

These rates determine not just the deterministic descent but also the scaling of fluctuations and higher-order corrections, often with central limit theorems or moderate deviation principles (Sagitov et al., 2016, Bansaye et al., 2015, Ojeda et al., 12 Jan 2025).

4. Phase Transitions and Universality Classes

Several processes display sharp phase transitions separating regions where CDFI holds from those where it fails:

• Fragmentation-coalescence (extreme shattering): For the Kingman coalescent with fragmentation rate $\lambda$, CDFI holds iff $\lambda < c/2$; at the boundary $\lambda = c/2$, the system is critical and does not come down (Kyprianou et al., 2016).
• Simple EFC processes: Threshold is set by the parameter $\theta$: CDFI if $\theta < 1$, stays infinite if $\theta>1$ (Foucart, 2016). For regular variation, thresholds can be explicit: e.g. if $\Phi(n)\sim d n^{1+\beta}$, $\overline\mu(n)\sim \lambda n^{-\alpha}$, then $\beta = 1-\alpha$ is critical.

These phase diagrams are robust across classes—analogous transitions occur in general coordinated particle systems, nested coalescents, and processes with generalized “infinite dispersion” mechanisms (Sreedhar, 16 Jun 2025, Benítez et al., 2018, Foucart, 2016).

5. Excursion Theory, Entrance Laws, and Pathwise Structure

The analytic structure of excursions from infinity is accessible in regimes exhibiting CDFI. For instance, in the fast fragmentation–coalescence process ($\lambda < c/2$), the local time at $0$ for $M(t) = 1/N(t)$ has a pure-jump subordinator inverse; the stationary law is Beta–Geometric, and explicit entrance laws and hitting times are computable: $\lim_{t\downarrow0} t\,N(t) = \frac{2}{c}$ from typical excursions (Kyprianou et al., 2016). The Hausdorff dimension of the set $\{t : N(t) = \infty\}$ is exactly the phase parameter $\theta$.

In metric settings, such as $\Lambda$-coalescent trees, CDFI is equivalent to the compactness of the associated metric measure space in the Gromov-weak topology. Failure to come down from infinity results in noncompact (and not even locally compact) limiting spaces (Biehler et al., 2011).

6. Generalizations and Future Directions

Current research extends CDFI results to multitype and spatially structured models (nested and coordinated coalescents, Fleming–Viot processes with initial infinite support), time-inhomogeneous transition rates, and SPDEs or field-theoretic limits (e.g. dynamic $\Phi^4_3$ models (Mourrat et al., 2016)).

Open questions include:

• CDFI for broad classes of exchangeable fragmentation–coalescence with arbitrary dislocation laws.
• Coupling genealogical or metric properties (e.g. fractal dimension of the zero set when $\theta<1$) to biologically interpretable quantities.
• Extensions to interacting diffusions and multiple interacting species, particularly in non-Lipschitz or non-expansive settings (Bansaye, 2015).
• Excursion theory and scaling at critical phase boundaries (Kyprianou et al., 2016, Foucart, 2016).

7. Illustrative Models and Table of Criteria

Process/Class	CDFI Criterion	Speed/Scaling Function
Kingman coalescent	Always (binary mergers at rate $c>0$)	$v(t) = 2/c t$
$\Lambda$-coalescent	$\sum_b \gamma_b^{-1} < \infty$	$v(t)$ via $\int_{v(t)}^\infty dq/\psi(q)=t$
Simple EFC	$\theta^\star < 1$	Finer scaling via $\Phi(n)$, fragmentation tail
Birth–death ($\lambda_n$,$\mu_n$)	$S=\sum_n \pi_n < \infty$	$v(t)$ from $E_\infty[T_n]$ inversion
Fragmentation–coalescence ($\lambda$,$c$)	$\lambda<c/2$	$tN(t)\to 2/c$
CB process (competition)	$\int_{z_0}^\infty dy/g(y)<\infty$	Solution of $dz/dt=-g(z)$
CB process (drift-interaction)	Lyapunov/test function method, drift $I(\cdot)$	Solution to associated ODE

• “A phase transition in excursions from infinity of the ‘fast’ fragmentation-coalescence process” (Kyprianou et al., 2016).
• “On the coming down from infinity of coalescing Brownian motions” (Barnes et al., 2022).
• “Time-changed spectrally positive Lévy processes starting from infinity” (Foucart et al., 2019).
• “The Nested Kingman Coalescent: Speed of Coming Down from Infinity” (Benítez et al., 2018).
• “Compact metric measure spaces and Lambda-coalescents coming down from infinity” (Biehler et al., 2011).
• “Extinction and coming down from infinity of CB-processes with competition in a Lévy environment” (Leman et al., 2018).
• “How do birth and death processes come down from infinity?” (Bansaye et al., 2013).
• “Speed of coming down from infinity for birth and death processes” (Bansaye et al., 2015).
• “Coming down from infinity for coordinated particle systems” (Sreedhar, 16 Jun 2025).
• “The hydrodynamic limit of beta coalescents that come down from infinity” (Miller et al., 2016).
• “A phase transition in the coming down from infinity of simple exchangeable fragmentation-coagulation processes” (Foucart, 2016).
• “Diffusions from Infinity” (Bansaye et al., 2017).
• “On the speed of coming down from infinity for (sub)critical branching processes with pairwise interactions” (Ojeda et al., 12 Jan 2025).
• “On the coming down from infinity of continuous-state branching processes with drift-interaction” (Rebotier, 7 Oct 2025).
• “Limit theorems for pure death processes coming down from infinity” (Sagitov et al., 2016).
• “Speed of coming down from infinity for $\Lambda$-Fleming-Viot initial support” (Liu et al., 8 Jun 2025).
• “Approximation of stochastic processes by non-expansive flows and coming down from infinity” (Bansaye, 2015).
• “Critical branching as a pure death process coming down from infinity” (Sagitov, 2021).
• “The dynamic $Φ^4_3$ model comes down from infinity” (Mourrat et al., 2016).