--- title: Yaglom Limit in Markov Processes url: https://www.emergentmind.com/topics/yaglom-limit type: topic --- # Yaglom Limit in Markov Processes The Yaglom limit is the limiting conditional law of an absorbed Markov process given non-absorption. In its classical form one asks whether, for each initial state $x$, the conditional distribution $\mathbb{P}_x(X_t\in A\mid \tau>t)$ in continuous time or $\mathbb{P}_x(X_n\in A\mid \tau>n)$ in discrete time converges as the conditioning horizon tends to infinity; in self-similar or unbounded settings, the relevant limit may instead involve a deterministic spatial rescaling of the process before conditioning [1709.06610] [1110.4795] [1612.03548]. Across the literature, Yaglom limits sit at the intersection of quasi-stationarity, spectral theory, Doob $h$-transforms, Martin boundary theory, excursion theory, and scaling limits, and they provide a precise description of the long-time behavior of systems that are certain to be absorbed but have not yet been absorbed. ## 1. Definitions and canonical formulations For a substochastic Markov kernel $K$ on a countable state space $S$, with absorption upon exit from $S$ and absorption time $\tau=\inf\{n\ge 0:X_n\notin S\}$, a Yaglom limit is a probability distribution $\nu$ such that $$ \lim_{n\to\infty}\mathbb{P}_x(X_n\in A\mid \tau>n)=\nu(A), $$ or equivalently $$ \frac{K^n(x,y)}{K^n(x,S)}\longrightarrow T_x(y), $$ with $T_x$ a probability measure on $S$ [1709.06610]. In continuous time, the analogous formulation is $$ \lim_{t\to\infty}\mathbb{P}_x(X_t\in A\mid \tau>t)=\nu(A). $$ When such a limit exists, it is a quasi-stationary distribution in the sense that conditioning at any fixed later time preserves the same law. A quasi-stationary distribution is a probability measure $\alpha$ or $\nu$ satisfying conditional invariance. In diffusion language, one writes $$ \alpha=P_\alpha(X_t\in\cdot\mid \tau_0>t), $$ while for substochastic kernels the same property is encoded by a left-eigenvector relation $$ \nu K=\rho\,\nu, $$ with $\rho\in(0,1]$ the relevant spectral radius parameter [1911.03693] [1709.06610]. In bounded-state or compact settings this often leads to an ordinary quasi-stationary regime with exponential survival, but several papers emphasize that in unbounded domains the Yaglom limit may require renormalization rather than a stationary law on the original scale. For self-similar processes, a Yaglom limit may take the rescaled form $$ \mathbb{P}_1\!\left(\frac{X_t}{g(t)}\in\cdot\mid tt\right)=\mu(A), $$ which is a rescaled conditional limit rather than a quasi-stationary law on the original space [1612.03548]. ## 2. Structural mechanisms behind Yaglom limits A central mechanism is the spectral one. For killed diffusions or killed semigroups, the Yaglom limit is often governed by the principal eigenvalue and associated positive eigenfunction. In Brownian motion with drift $-r$ on $(0,\infty)$ absorbed at $0$, the principal eigenvalue is $\lambda_0=r^2/2$, and the principal eigenfunction is proportional to $x\mapsto x e^{r x}$; the paper fixes $$ \eta(x)=\frac{1}{r^2}x e^{r x}. $$ The Yaglom limit then has explicit density $$ \alpha_{Yaglom}(dx)=r^2 x e^{-r x}\,dx, $$ and under that law the survival probability decays as $e^{-\lambda_0 t}$ [1911.03693]. The same spectral principle appears in more general standard processes with no negative jumps. There the resolvent is written in terms of scale functions $W^{(q)}$ and $Z^{(q)}$, and the spectrum of the generator on $(0,\infty)$ killed at $0$ is characterized by the zeros of the entire function $q\mapsto Z^{(q)}(0)$. Under the entrance boundary condition at $+\infty$, the unique quasi-stationary distribution is $$ \nu_{\lambda_0}(dx)=\lambda_0\,W^{(-\lambda_0)}(0,x)\,m(dx), $$ and $$ \lim_{t\to\infty}e^{\lambda_0 t}p_t f(x)=\frac{Z^{(-\lambda_0)}(x)}{\lambda_0\rho}\,\nu_{\lambda_0}(f) $$ in total variation form [2410.15447]. A second structural mechanism is the Doob $h$-transform, which produces the conditioned process that never gets absorbed. For $R$-transient chains with spectral parameter $\rho=1/R$, the transformed kernel is $$ K^{(h)}(x,y)=\frac{1}{\rho}\frac{h(y)}{h(x)}K(x,y). $$ This $h$-transform is central both to the description of conditioned trajectories and to the decomposition of Yaglom limits when the Martin boundary is non-trivial [1709.06610]. In the Brownian-with-drift model, the corresponding $Q$-process is the Bessel-$3$ diffusion $$ dY_t=dB_t+\frac{1}{Y_t}\,dt, $$ which is obtained from the killed diffusion by Doob transform with $h=\eta$ and represents conditioning on eternal survival [1911.03693]. A third mechanism is boundary theory. For $R$-transient chains with non-trivial $\rho$-Martin entrance boundary, the set of $\rho$-invariant probabilities is the convex hull of extremal entrance-boundary measures, and the Yaglom limit may be a boundary-dependent mixture rather than a single universal law [1709.06610]. For a single-exit-state continuous-time chain, the relevant structure is excursion theory: the absorption time is represented through the inverse local time at the exit state, the inverse local time is a subordinator, and the minimal quasi-stationary distribution is expressed directly through the excursion law [2406.06188]. ## 3. Representative model classes In one-dimensional killed diffusions, the Yaglom limit is often explicit. For Brownian motion with constant drift $-r$ on $(0,\infty)$ absorbed at $0$, $$ X_t=X_0+B_t-r t,\qquad \tau_0=\inf\{t\ge 0:X_t=0\}, $$ the unique Yaglom limit attracting every Dirac initial law has density $r^2 x e^{-r x}\,dx$ [1911.03693]. The same model also exhibits infinitely many quasi-stationary distributions, so uniqueness of the Yaglom limit is stronger than mere existence of quasi-stationarity. In branching settings, the classical exponential Yaglom law persists but the normalization becomes model-dependent. For a critical Galton–Watson process in varying environment, with $$ \mu_n=\prod_{i=1}^n f_i'(1),\qquad \nu_n=\frac{f_n''(1)}{f_n'(1)^2},\qquad \rho_{0,n}=\sum_{k=0}^{n-1}\frac{\nu_{k+1}}{\mu_k}, $$ one has, under regularity and criticality, $$ \frac{Z_n}{b_n}\,\Big|\,\{Z_n>0\}\Rightarrow \mathrm{Exp}(1), \qquad b_n\sim \frac{\mu_n\rho_{0,n}}{2}, $$ and an equivalent normalization is $a_n=\mu_n\rho_{0,n}/2$ [2005.10186]. In critical neutron transport, the analogous result is $$ \mathrm{Law}\!\left(\frac{\langle f,X_t\rangle}{t}\,\Big|\,\langle 1,X_t\rangle>0\right)\to \mathbf e_{c(f)}, \qquad c(f)=\frac{2}{\langle \tilde\phi,f\rangle \Sigma}, $$ with survival probability asymptotic $2\phi(x)/(\Sigma t)$ [2103.02237]. For subcritical branching Markov chains, the Yaglom limit is a law on nonzero finite point measures. Under assumptions denoted (C1)–(C3), there exists a unique probability measure $Q_{\min}$ on $N^\circ(\mathbb R)$ such that $$ \mathbb P_\mu(Z_n\in\cdot\mid N_n>0)\Rightarrow Q_{\min}, $$ for every $\mu\in N^\circ(\mathbb R)$, and $Q_{\min}$ is the unique quasi-stationary distribution with eigenvalue $m$ [2405.04284]. The same paper gives an explicit integral representation of all quasi-stationary distributions with eigenvalue $m^\alpha$, $\alpha\in(0,1)$, in terms of the probability generating functional of $Q_{\min}$. For branching Brownian motion with absorption at slightly subcritical drift $\rho>\sqrt{2}$, the conditioned law of the entire configuration converges to a quasi-stationary distribution $D^\rho$, and $$ \mathbb P_{D^\rho}(\zeta>t)=e^{-(\rho^2/2-1)t}. $$ This Yaglom law is minimal and universal over nonzero initial configurations [2409.08789]. ## 4. Rescaled Yaglom limits and self-similar geometry In cone-like geometries, the correct Yaglom limit lives on a rescaled state space. For isotropic $\alpha$-stable Lévy processes killed on a Lipschitz cone $\Gamma$, there exists a probability measure $\mu$ such that $$ \mathbb P_x\!\left(t^{-1/\alpha}X_t\in A\mid \tau_\Gamma>t\right)\to \mu(A). $$ The key input is the Martin kernel at the vertex, $$ M(y)=\lim_{x\to 0}\frac{G_\Gamma(x,y)}{G_\Gamma(x,1)}, $$ which is homogeneous of degree $\beta\in(0,\alpha)$, and the survival probability satisfies $$ \mathbb P_x(\tau_\Gamma>t)\sim C_1 M(x)t^{-\beta/\alpha}. $$ The associated entrance-law density $n_t$ obeys $$ n_t(y)=t^{-(d+\beta)/\alpha}n_1(t^{-1/\alpha}y) $$ [1612.03548]. This cone theory extends beyond the isotropic symmetric case. For non-symmetric strictly $\alpha$-stable processes with spherical density bounded and bounded away from zero, killed in a $\kappa$-fat cone, there are two Martin kernels, $M_\Gamma$ and $\widehat M_\Gamma$, with homogeneity exponents $\beta$ and $\widehat\beta$, and the rescaled conditional law converges to a measure with density proportional to $\phi(y)\widehat M_\Gamma(y)$, where $\phi$ is the stationary density of an associated Ornstein–Uhlenbeck-type semigroup [2310.08570]. A notable feature is that $\beta$ and $\widehat\beta$ can differ, so the forward and dual boundary behaviors need not coincide. For unimodal Lévy processes sufficiently close to isotropic $\alpha$-stable, the cone Yaglom limit is universal after the correct process-dependent normalization. If $$ X_t^s=\psi^{-1}(1/s)X_{st}, $$ then under assumptions A1–A3 and for a Lipschitz cone $\Gamma$, $$ \lim_{s\to\infty}\mathbb P_x\!\left(\psi^{-1}(1/s)X_s\in A\mid \tau_\Gamma>s\right)=\mu^\alpha(A), $$ where $\mu^\alpha$ is exactly the $\alpha$-stable cone Yaglom limit [2110.00873]. The universality concerns the limiting measure, not the rescaling map $\psi^{-1}(1/s)$. For positive self-similar Markov processes that hit $0$ in finite time, the existence of a Yaglom limit is equivalent to the extinction time being in the domain of attraction of an extreme-value law. The limit is then characterized by a multiplicative factorization involving the exponential functional $$ I=\int_0^\zeta e^{\alpha \xi_s}\,ds $$ of the Lamperti Lévy process: $RI\stackrel d=e$ in the Gumbel case, $R_\gamma I\stackrel d=B_\gamma$ in the Weibull case, and $J_\gamma I\stackrel d=P_\gamma$ in the Fréchet case [1110.4795]. ## 5. Quantitative convergence and rates Yaglom theory is not only asymptotic; several recent papers quantify convergence. For Brownian motion with drift $-r$ on $(0,\infty)$ absorbed at $0$, if the initial law $\mu$ satisfies $$ \int_0^\infty x^3 e^{r x}\,\mu(dx)<\infty, $$ then $$ 0<\liminf_{t\to\infty} t\,W_1\!\left(\mathbb P_\mu(X_t\in\cdot\mid \tau_0>t),\alpha_{Yaglom}\right) \le \limsup_{t\to\infty} t\,W_1\!\left(\mathbb P_\mu(X_t\in\cdot\mid \tau_0>t),\alpha_{Yaglom}\right) <\infty, $$ so the sharp rate is $1/t$ [1911.03693]. The same paper states that the same $1/t$ bounds hold if $W_1$ is replaced by total variation or Kolmogorov distance, and it proves an analogous $1/t$ rate for convergence to the Bessel-$3$ $Q$-process at fixed observation time. In critical Galton–Watson processes in varying environment, the convergence to the exponential Yaglom law can also be quantified in Wasserstein distance. Under the mild third-moment condition $$ f_n'''(1)\le c\,f_n''(1)\bigl(1+f_n'(1)\bigr) $$ and criticality, the bound $$ W_1\!\left(\mathrm{Law}\!\left(\frac{Z_n}{b_n}\,\Big|\,Z_n>0\right),\mathrm{Exp}(1)\right) \le C\left(\frac{1}{\mu_n\rho_{0,n}+r_n/\rho_{0,n}}\right) $$ holds, with explicit $r_n$ [2305.11801]. Under further regularity one obtains logarithmically corrected bounds of order $(\mu_n\rho_{0,n})^{-1}$. The opposite regime also occurs. For standard processes with no negative jumps under an entrance boundary at $+\infty$, once the strong Feller property is assumed, the killed semigroup is compact on $C_\infty$, the spectrum is discrete, and the Yaglom limit is approached exponentially fast: $$ \sup_{\|f\|_\infty\le 1}\big\|e^{\lambda_0 t}p_t f-\pi_{\lambda_0}f\big\|_\infty \le M_\delta e^{-\delta t}, \qquad \delta\in(0,\lambda_1-\lambda_0). $$ Consequently the conditioned law converges exponentially in total variation to $\nu_{\lambda_0}$ [2410.15447]. This contrast with the polynomial $1/t$ regime shows that convergence speed is model-specific rather than universal. ## 6. Non-uniqueness, boundary effects, and dependence on the initial state A common misconception is that a Yaglom limit, when it exists, must be unique and independent of the initial state. Several papers show that this is false. For $R$-transient chains with non-trivial $\rho$-Martin entrance boundary, the space of $\rho$-invariant quasi-stationary distributions can have multiple extremal points, and the Yaglom limit may depend on the initial condition through the exit distribution of the Doob $h$-transform [1709.06610]. An explicit example is the “hub-and-two-spoke” chain on $\mathbb Z$ with killing only at $0$. There the Yaglom limit exists for every starting state $x$, but it is $$ \pi_x(y)= \begin{cases} \displaystyle \frac{1-\rho}{2a}\Bigl(1+|y|+\xi(x)y\Bigr)\left(\sqrt{\frac{b}{a}}\right)^{|y|}, & y\neq 0,\\[1ex] \displaystyle \frac{1-\rho}{a}, & y=0, \end{cases} $$ with $\xi(x)=x/(1+|x|)$, so different starting states produce different quasi-stationary limits [1709.07578]. The dependence is explained by a two-point $\rho$-Martin entrance boundary $\{-\infty,+\infty\}$ and the resulting convex family of $\rho$-invariant measures. The same boundary picture can also lead to non-existence rather than multiplicity. The general theory in the $R$-transient nearest-neighbor setting requires the Jacka–Roberts condition; when it fails, as in Kesten’s counterexample, the Yaglom limit may fail to exist even though ratio limits and entrance-boundary extremals are present [1709.06610]. By contrast, in the single-exit-state setting the excursion representation produces a minimal quasi-stationary distribution explicitly and yields existence of the Yaglom limit under the stated threshold condition $\lambda<\lambda_c$ [2406.06188]. ## 7. Applications, computation, and current directions The concept has developed far beyond classical branching and birth–death models. In critical neutron transport with non-local branching, the Yaglom limit describes the conditional asymptotic mass of the particle system: $$ \lim_{t\to\infty}\mathbb E_{\delta_x}\!\left[\exp\!\left(-\theta\frac{\langle f,X_t\rangle}{t}\right)\Bigm|\zeta>t\right] = \frac{1}{1+\langle \tilde\phi,f\rangle \Sigma\theta/2}, $$ so the limit law is exponential with rate $2/(\langle\tilde\phi,f\rangle\Sigma)$ [2103.02237]. In stochastic fluid models, the Yaglom limit is computed through the singularity $s^*$ of the matrix $\Psi(s)$; a square-root expansion $$ \Psi(s)=\Psi(s^*)-B(s^*)\sqrt{s-s^*}+o(\sqrt{s-s^*}) $$ drives the limiting conditional distribution and its uniqueness [1908.10827]. For continuous-state branching processes in Brownian random environment, the subcritical Yaglom limit exists in the weakly, intermediately, and strongly subcritical regimes, and its Laplace transform is given explicitly in terms of Kummer confluent hypergeometric functions $U$. In all three regimes the limit is independent of the initial state $z>0$ [2605.27949]. This provides a continuous-state analogue of a question that remains delicate in weakly subcritical Galton–Watson processes in random environment. On the computational side, Fleming–Viot-type interacting particle systems provide a numerical approximation to Yaglom limits for absorbed diffusions with unbounded drift on unbounded domains. On bounded truncations $U_m$, the empirical stationary measure of the $N$-particle system converges to the Yaglom limit $\nu_m$, and then $\nu_m\Rightarrow \nu_\infty$ as $m\to\infty$, yielding $$ \lim_{m\to\infty}\lim_{N\to\infty}X_{m,N}=\nu_\infty $$ in the weak topology [1005.1530]. This suggests a practical route when direct spectral computation of the quasi-stationary law is infeasible. Taken together, these results show that the Yaglom limit is not a single theorem but a family of asymptotic phenomena. Depending on geometry, scaling, branching structure, and boundary behavior, it may be an explicit quasi-stationary distribution, a rescaled conditional law, a mixture indexed by boundary points, an exponential law for a suitably normalized observable, or a numerically approximable invariant object for a conditioned particle system. Source: https://www.emergentmind.com/topics/yaglom-limit