---
title: Yaglom Type Limit Theorem
url: https://www.emergentmind.com/topics/yaglom-type-limit-theorem
type: topic
---

# Yaglom Type Limit Theorem

The Yaglom Type Limit Theorem describes the asymptotic, conditioned behavior of Markovian and stochastic processes that are absorbed (killed) at a boundary or upon exit from a domain: specifically, it addresses the weak limit of the process’s distribution, conditioned on non-absorption as time tends to infinity, often after a suitable rescaling. This limit—termed the Yaglom limit—typically manifests as a quasi-stationary distribution (QSD), characterizing the long-run profile of trajectories that avoid absorption indefinitely. The formal structure, generality, and applicability of Yaglom-type limits span a broad array of settings, including non-symmetric stable Lévy processes in cones, Galton-Watson processes, Markov chains with various boundary conditions, and infinite-dimensional measure-valued processes.

## 1. Foundational Structure for Non-Symmetric Stable Processes

The general Yaglom-type limit is exemplified by non-symmetric, strictly $\alpha$-stable Lévy processes $X$ in $\mathbb{R}^d$ ($0<\alpha<2$), with Lévy measure $\nu(dz)=|z|^{-d-\alpha}\lambda(z/|z|)\,dz$ and bounded, strictly positive spherical density $\lambda$. Let $\Gamma\subset\mathbb{R}^d$ denote a closed, scale-invariant $\kappa$-fat cone (satisfying the interior-ball condition relative to its boundary). Let $\tau_\Gamma = \inf\{t>0:X_t\notin\Gamma\}$ be the exit time, and $p^\Gamma(t,x,y)$ the corresponding Dirichlet heat kernel. The minimal positive harmonic function in $\Gamma$, the Martin kernel $M_\Gamma(x)$, is uniquely characterized by $M_\Gamma(kx)=k^\beta M_\Gamma(x)$ for some $\beta\in[0,\alpha)$. 

The central result asserts that for any probability measure $\gamma$ on $\Gamma$ with finite $(1+|x|)^\alpha$-moment, there exists a uniquely determined measure $\mu$ on $\Gamma$, with density 
\[
\mu(dy) = C^{-1}\,\phi(y) M_\Gamma(y)\,dy, \quad C = \int_\Gamma \phi(y) M_\Gamma(y)\,dy,
\]
where $\phi$ is the unique invariant density for the normalized Ornstein–Uhlenbeck semigroup associated with the conditioned kernels $\rho_t(x,y) = p^\Gamma(t,x,y) / [M_\Gamma(x)M_\Gamma(y)]$. For every Borel set $A\subset\Gamma$,
\[
\lim_{t\to\infty} P_\gamma\left(t^{-1/\alpha} X_t \in A\,\big|\,\tau_\Gamma>t\right) = \mu(A),
\]
uniformly in the starting distribution $\gamma$ and, in particular, for all $x\in\Gamma$ [2310.08570].

## 2. Key Kernel Estimates and Martin Boundary Structure

For non-symmetric strictly $\alpha$-stable processes killed on exit from $\kappa$-fat cones, the approach is predicated on sharp two-sided Dirichlet heat-kernel estimates:
\[
c_1 P_x(\tau_\Gamma>t)\,p(t,x,y)\,P_y(\tau_\Gamma>t) \leq p^\Gamma(t,x,y) \leq c_2 P_x(\tau_\Gamma>t)\,p(t,x,y)\,P_y(\tau_\Gamma>t),
\]
with $p(t,x,y)$ the free heat kernel, and $c_1, c_2$ depending only on $(d,\alpha,\lambda,\kappa)$.

The Martin kernel $M_\Gamma(x)$ is constructed as the minimal non-negative, regular $\alpha$-harmonic function vanishing on $\partial\Gamma$, normalized at a basepoint $e\in\Gamma$, and exhibits scale-homogeneity $M_\Gamma(kx)=k^\beta M_\Gamma(x)$. The Yaglom density $\mu$ is then re-expressed using the invariant density $\phi$ via
\[
\phi(y) \approx (1+|y|)^{-d-\alpha} \cdot \frac{P_y(\tau_\Gamma>1)}{M_\Gamma(y)},
\]
allowing explicit characterization of the quasi-stationary profile [2310.08570].

## 3. Survival Asymptotics and Quasi-Stationary Law

Spatial asymptotics for the survival probability are connected to the Martin kernel,
\[
P_x(\tau_\Gamma > t) \sim C M_\Gamma(x) t^{-\beta/\alpha}\quad\text{as } t\to\infty,
\]
with normalization constant $C = \int_\Gamma \phi(y) M_\Gamma(y)\,dy$. The Yaglom limit law $\mu$ is uniquely quasi-stationary: evolving under the killed process, starting from $\mu$ ensures
\[
P_\mu\left(X_t\in A\,\big|\,\tau_\Gamma>t\right)=\mu(A)
\]
for all Borel $A\subset\Gamma$ and all $t>0$ [2310.08570].

## 4. Proof Strategy and Connections

The proof adapts a Varopoulos-type kernel factorization, tailored to the non-symmetric setting, always pairing $X$ with its dual. The approach leverages:
- Global Dirichlet kernel factorization using both $X$ and $\widehat{X}_t = -X_t$,
- Identification and scaling properties of the Martin kernel,
- Renewal-type and compactness (Dini-type) arguments for existence of limiting measures,
- Construction of the Ornstein–Uhlenbeck semigroup and extraction of its unique stationary law through normalization,
- The final compactness step shows convergence of the normalized kernels $\rho_t$ toward the invariant density $\phi(y)$ as $t \to \infty$ (or $x \to 0$ under scaling), yielding the explicit Yaglom limit.

Both the existence of factorized kernel bounds and uniqueness of the invariant law rely critically on the geometry of $\Gamma$ (via the $\kappa$-fat condition), scaling properties, and the boundedness of $\lambda$.

## 5. General Context, Universality, and Related Results

Yaglom-type limits for absorbed stable processes in cones form part of a broader theoretical landscape encompassing:
- Symmetric and non-symmetric stable processes [1612.03548, 2310.08570],
- General unimodal Lévy processes (establishing universality of the limit profile under suitably regular scaling and regular variation assumptions) [2110.00873],
- Self-similar Markov processes, connecting the existence of Yaglom limits to the extreme-value domains of the extinction time and its exponential functional representations [1110.4795].

Notably, extensions and parallels also arise in:
- Critical and subcritical branching processes and multitype branching systems,
- Discrete and continuous-time Markov chains conditioned to non-absorption,
- Diffusions with unbounded drift via particle system approximation schemes.

Further, the Yaglom limit for non-symmetric stable processes in cones connects to boundary Harnack principles, entrance laws from the cone vertex, exact survival probability asymptotics, and modern potential-theoretic approaches [2310.08570, 1612.03548, 2110.00873].

## 6. Extensions and Broader Implications

The structural insight that the Yaglom limit density in a $\kappa$-fat (possibly non-symmetric) cone is given explicitly in terms of the product of the Martin kernel and the stationary density of a conditioned Ornstein–Uhlenbeck operator offers a versatile framework, applicable to a broad class of stable processes and geometric domains. The result is robust under non-isotropy and non-symmetry, contingent primarily on boundedness conditions for the Lévy spectral measure and the $\kappa$-fatness geometry. This framework is fundamental for understanding the spatial asymptotics of surviving paths, the law of large-time conditioned processes, and the analytic structure of quasi-stationary measures.

Moreover, the association with the Martin boundary and harmonic function theory provides a robust analytic toolkit for characterizing conditioned long-time limits of absorbed Markovian dynamics in both finite and infinite dimensional settings.

**References:**  
[2310.08570] Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit  
[1612.03548] Yaglom limit for stable processes in cones  
[2110.00873] Yaglom limit for unimodal Lévy processes  
[1110.4795] Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

Source: https://www.emergentmind.com/topics/yaglom-type-limit-theorem