Yaglom Limit Theorems in Stochastic Processes

• Yaglom Type Limit Theorems describe the asymptotic behavior of stochastic processes conditioned on survival, unifying spectral, potential, and quasi-stationary analyses.
• They employ techniques such as spectral theory, excursion methods, and nonlocal operator analysis to derive universal limit distributions in diverse mathematical models.
• Applications span branching processes, Lévy flights, and random media, providing crucial insights into metastability, rare-event probabilities, and long-time conditioned dynamics.

A Yaglom type limit theorem describes the asymptotic conditional law of a stochastic process, typically one that is absorbed or killed, given that it has not yet reached its absorbing state. For a wide range of models including finite and infinite-dimensional Markov processes, Lévy processes, and branching processes, the law of the system, when conditioned on non-extinction or non-absorption and properly normalized (or rescaled), converges to a non-trivial, often universal, limit distribution. These results form the backbone of modern quasi-stationary analysis, linking probabilistic, spectral, and potential-theoretic methodologies with applications in random media, evolutionary dynamics, and nonlocal PDEs.

1. General Concept and Classical Setting

The Yaglom limit encapsulates the limiting behavior of a process absorbed at a boundary (or killed) and conditioned on survival up to large times. This notion originated in the theory of critical Galton–Watson branching processes, where the conditional population size, upon survival, converges in law to a nontrivial quasi-stationary distribution (QSD). For a Markov process $(X_t)_{t\geq 0}$ with absorbing state $\partial$ and extinction/killing time $\tau_\partial$,

$$\nu^*(A) = \lim_{t \to \infty} P_x(X_t \in A \mid \tau_\partial > t)$$

defines (when it exists and is independent of $x$) the Yaglom limit, i.e., the minimal QSD.

In critical branching and branching Brownian motion, the Yaglom limit describes the size or configuration of a system, given atypical long survival. Modern formulations encompass measure-valued, infinite-dimensional, or spatially inhomogeneous dynamics, often leveraging spectral theory, excursion measures, or asymptotic analysis.

2. Matrix and Spectral Formulations

In discrete (lattice) and continuum models, computation of Yaglom limits is linked to spectral properties and determinants of relevant differential or difference operators. For discrete Sturm–Liouville-type problems, the Gel'fand–Yaglom formula provides an efficient means of computing determinants of second-order operators without explicit eigenvalues, connecting to classic spectral zeta function regularization. For instance, in the presence of Dirichlet boundary conditions, the determinant is given by the value $y(v+1,0)$ of the fundamental solution of the associated recurrence. Extensions using matrix (phase-space) approaches allow for boundary condition flexibility and enable trace formulation, illuminating symplectic invariance and links to orthogonal polynomial theory (Dowker, 2011).

For processes with generator $A$ (possibly a nonlocal operator, e.g., stable processes), the spectral picture often reduces the long-time, conditioned behavior to the dominance of a principal eigenvector or positive solution of $A f + \lambda_0 f = 0$, with normalization dictated by survival rates and boundary behavior. In the context of continuous-state branching or Lévy processes in cones, the existence and uniqueness of the Yaglom limit are characterized via the spectrum of $A$ (or entire functions constructed from scale or Green functions) (Yamato, 20 Oct 2024).

3. Yaglom Limits in Spatial and Non-Local Models

High-dimensional and non-local Markov processes—including isotropic or non-symmetric $\alpha$-stable processes, Lévy flights, and general jump processes—extend the scope of Yaglom limits. A central result is that for an isotropic $\alpha$-stable process $X_t$ in a Lipschitz cone $T \subset \mathbb{R}^d$, conditioned on $T_t > t$ (not exiting the cone by time $t$) and after rescaling by $t^{-1/\alpha}$, the law converges to a universal measure $\mu$ on $T$,

$$\lim_{t \to \infty} P_x\big(t^{-1/\alpha} X_t \in A \mid T_t > t\big) = \mu(A)$$

where $M(x)$, the Martin kernel, describes spatial asymptotics and entrance laws from the boundary (Bogdan et al., 2016). Crucially, this limit can be described in terms of harmonic functions of the generator (fractional Laplacian) and is robust to perturbations of the Lévy measure, establishing universality for broad classes of unimodal (not necessarily self-similar) Lévy processes (Armstrong et al., 2021). The theory further generalizes to non-symmetric $\alpha$-stable processes in $\kappa$-fat cones, where the scaling and limiting distribution are controlled by the Martin kernel's homogeneity exponent $\beta$, which may depend on anisotropy (Leżaj, 2023).

4. Branching and Random Environment Models

Yaglom-type limit theorems in branching processes typically analyze the system's structure, given persistence. In finite or Markovian environments, the normalization required involves additive Markov walks $S_n$, and the joint limiting laws of $(\log Z_n/\sqrt{n}, X_n)$ (size and environment) converge to non-degenerate universal (Rayleigh-type) limits, characterized by harmonic functions and invariant measures of the environment chain (Grama et al., 20 Dec 2024). Similar universality phenomena and scaling exponents (e.g., $\epsilon^{-1/3}$ scaling as drift approaches criticality) have been identified in branching Brownian motion with absorption (Berestycki et al., 13 Sep 2024).

In infinite-dimensional, trait-structured, or measure-valued models, Feynman–Kac semigroups, spectral gaps, and Doob h-transforms yield quasi-stationary laws (Yaglom limits) for the normalized system conditioned on non-extinction, with normalization rates and moment recursions determined by the spectral gap and principal eigenfunction (Collet et al., 6 Aug 2025).

For decomposable multitype branching in random environments, environmental stochasticity dramatically modifies survival probabilities and asymptotics (e.g., $1/\log n$ decay, as opposed to polynomial) and spreads survival among types rather than “collapse” to the highest type found in constant environment cases (Vatutin et al., 2014).

5. Analytical and Excursion-Theoretic Approaches

Modern analyses of Yaglom limits often exploit local time, excursion theory, and large deviations for Markov and birth–death processes. For chains with absorption only from a single state, representation by excursions and the associated subordinator enables explicit computation of the QSD in terms of Laplace transforms and excursion length distributions; the exponential decay rate $\theta^*$ is determined as the solution to $\psi(\theta^*) = \lambda$ for the Laplace exponent $\psi$ of the subordinator (Cerf, 10 Jun 2024).

Exponential convergence (or polynomial, in cases such as Brownian motion with drift (Oçafrain, 2019)) to the Yaglom limit is established under spectral gap assumptions. The spectral gap is characterized via scale functions and the compactness of the Feller semigroup, with the spectrum determined by zeros of entire functions $q\mapsto Z^{(q)}(0)$ (Yamato, 20 Oct 2024).

6. Stochastic Dynamics, Fluctuation Theory, and Prefactor Calculations

In stochastic dynamical systems and rare-event calculations, Yaglom-type asymptotics inform both the exponential scaling (instanton action) and the leading-order prefactor for observable distributions. Gel'fand–Yaglom-type recursive methods compute fluctuation determinants by reducing the path integral problem to a matrix Riccati equation for the covariance along the instanton trajectory, yielding closed-form corrections to the exponential law that governs the Yaglom limit in function space (Schorlepp et al., 2021). This machinery generalizes to high-dimensional and SPDE settings, enabling precise computation of rare event probabilities and distributional tails.

7. Universality, Minimality, and Martin Boundary Effects

A recurring structural aspect is the universality of the Yaglom limit: for a broad class of processes (asymptotically stable, within domain of attraction, etc.), the conditional law (appropriately normalized) converges to the same law as the stable model (Armstrong et al., 2021). Uniqueness and minimality of the QSD/Yaglom limit are typically guaranteed unless the Martin entrance boundary is nontrivial, in which case the limiting law may depend on the starting state (Foley et al., 2017, Foley et al., 2017). The explicit connection to minimal harmonic functions, the Martin boundary, and Doob h-transforms underpins much of the modern theory.

The following table summarizes key universal structural features across models:

Model Class	Scaling/Normalization	Limiting Law Structure
Stable processes in cone	$t^{-1/\alpha} X_t$	Law $\mu$ via Martin kernel
Branching, critical	$(\log Z_n)/\sqrt{n}$ (Markovian env.)	Rayleigh/exponential law
Markov chain, single exit	Excursion representation, rate $\theta^*$	QSD via excursion measure
Birth–death/trait-structured	Spectral (eigenfunction) normalization	QSD and Yaglom limit

Conclusion

Yaglom type limit theorems unify a diverse set of probabilistic phenomena, from finite branching systems to nonlocal Lévy dynamics and infinite-dimensional measure-valued models. The central insight is that, while extinction or absorption is certain, the process—when conditioned on improbable survival—exhibits robust, universal limiting behavior determined by intrinsic spectral (or boundary) data and, in many cases, is independent of microscopic details or initial conditions. These theorems are now foundational in understanding quasi-stationarity, metastability, and long-time conditioning in modern probability and its applications.