Yaglom Type Limit Theorem

• Yaglom Type Limit Theorem defines the long-term behavior of Markovian processes conditioned on non-absorption, yielding a quasi-stationary distribution.
• It uses sharp two-sided Dirichlet heat-kernel estimates and the Martin kernel framework in κ-fat cones to precisely capture survival asymptotics.
• The theorem underpins the extraction of an invariant density via a conditioned Ornstein–Uhlenbeck semigroup, linking kernel factorizations to quasi-stationary laws.

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 $x\in\Gamma$ (Leżaj, 2023).

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 (Leżaj, 2023).

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 $t>0$ (Leżaj, 2023).

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 (Bogdan et al., 2016, Leżaj, 2023),
• General unimodal Lévy processes (establishing universality of the limit profile under suitably regular scaling and regular variation assumptions) (Armstrong et al., 2021),
• Self-similar Markov processes, connecting the existence of Yaglom limits to the extreme-value domains of the extinction time and its exponential functional representations (Haas et al., 2011).

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 (Leżaj, 2023, Bogdan et al., 2016, Armstrong et al., 2021).

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:

(Leżaj, 2023) Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit (Bogdan et al., 2016) Yaglom limit for stable processes in cones (Armstrong et al., 2021) Yaglom limit for unimodal Lévy processes (Haas et al., 2011) Quasi-stationary distributions and Yaglom limits of self-similar Markov processes