---
title: Q-Process in Markov and Branching Systems
url: https://www.emergentmind.com/topics/q-process
type: topic
---

# Q-Process in Markov and Branching Systems

In probability theory, the **Q-process** is the Markov process obtained from an almost surely absorbed Markov process by conditioning on eternal survival, or equivalently by taking the limit of conditioning on non-absorption at a remote future time. For a process \((X_t)_{t\ge 0}\) on \(E\cup\{\partial\}\) with absorbing cemetery state \(\partial\) and absorption time \(\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}\), the Q-process is defined, when the limit exists, by
\[
Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.
\]
Its modern theory is inseparable from quasi-stationary distributions (QSDs), uniform exponential convergence of conditioned laws, and a Doob \(h\)-transform built from survival asymptotics [1404.1349, 1611.02473]. In branching-process literature, the same construction appears as conditioning on non-extinction far in the future, and several extensions condition on thresholds, non-absorbing sets, or large total progeny [1412.3322, 2203.16904].

## 1. Absorbed Markov processes and quasi-stationarity

The standard setting is a time-homogeneous Markov process \((X_t)_{t\ge 0}\) on \(E\cup\{\partial\}\), where \(\partial\) is absorbing in the sense that
\[
X_s=\partial \implies X_t=\partial \quad \forall t\ge s.
\]
The standing assumptions are that \(\tau_\partial<\infty\) almost surely under every \(P_x\), while \(P_x(t<\tau_\partial)>0\) for all \(x\in E\) and \(t\ge 0\). A probability measure \(\alpha\) on \(E\) is a quasi-stationary distribution if
\[
P_\alpha(X_t\in\cdot\mid t<\tau_\partial)=\alpha(\cdot),\qquad \forall t\ge 0.
\]
If \(X_0\sim\alpha\), the conditioned law is therefore invariant at every time. When \(\alpha\) is a QSD, there exists \(\lambda_0>0\) such that
\[
P_\alpha(t<\tau_\partial)=e^{-\lambda_0 t},\qquad e^{\lambda_0 t}\alpha P_t=\alpha.
\]
This exponential survival law identifies the decay rate governing the absorbed process and simultaneously supplies the spectral parameter entering the Q-process construction [1404.1349].

The Q-process is conceptually the “never absorbed” version of the killed process. It is not obtained by removing the absorbing state from the state space, but by a limiting reweighting of path probabilities. In that sense, the Q-process is the canonical conservative dynamics associated with an absorbed system once long-time survival is imposed. This perspective is central both in general Markov-process theory and in branching-process conditioning.

## 2. Existence criteria and exponential convergence to a unique QSD

A decisive characterization is given by Assumption (A): there exists a probability measure \(\nu\) on \(E\) such that

\[
P_x(X_{t_0}\in \cdot \mid t_0<\tau_\partial)\ge c_1\,\nu(\cdot)
\]
for some \(t_0,c_1>0\) and all \(x\in E\), and
\[
P_\nu(t<\tau_\partial)\ge c_2\,P_x(t<\tau_\partial)
\]
for some \(c_2>0\), all \(x\in E\), and all \(t\ge 0\). The first condition is a uniform minorization for the conditioned process; the second requires survival from \(\nu\) to be uniformly comparable to survival from any point. Equivalent variants, denoted (A′) and (A′′), are also available [1404.1349].

These conditions are equivalent to uniform exponential convergence of conditioned laws toward a unique quasi-stationary distribution. More precisely, the following are equivalent: Assumption (A); its equivalent variants; existence of a probability measure \(\alpha\) and constants \(C,\gamma>0\) such that
\[
\left\|P_\mu(X_t\in\cdot\mid t<\tau_\partial)-\alpha(\cdot)\right\|_{TV}\le Ce^{-\gamma t},\qquad \forall \mu,\ t\ge 0;
\]
the corresponding pointwise form for all initial states \(x\in E\); and the integrability condition
\[
\int_0^\infty \sup_{x\in E}
\left\|P_x(X_t\in\cdot\mid t<\tau_\partial)-\alpha(\cdot)\right\|_{TV}\,dt<\infty.
\]
Whenever these equivalent properties hold, \(\alpha\) is the unique QSD, convergence is exponential and uniform in the initial law, and one obtains the explicit estimate
\[
\left\|P_\mu(X_t\in\cdot\mid t<\tau_\partial)-\alpha(\cdot)\right\|_{TV}
\le 2(1-c_1c_2)^{\lfloor t/t_0\rfloor}.
\]
The same hypotheses also ensure existence and exponential ergodicity of the Q-process itself [1404.1349].

This equivalence is one of the structural results of the subject. It shows that the Q-process is not an auxiliary construction attached to a pre-existing QSD theory; rather, uniform QSD convergence, survival asymptotics, and long-time conditioning are different manifestations of the same probabilistic regime.

## 3. Doob \(h\)-transform structure of the Q-process

Under the preceding hypotheses, survival probabilities admit a uniform asymptotic renormalization through a positive eigenfunction
\[
\eta(x)=\lim_{t\to\infty}\frac{P_x(t<\tau_\partial)}{P_\alpha(t<\tau_\partial)}
=\lim_{t\to\infty}e^{\lambda_0 t}P_x(t<\tau_\partial),
\qquad \alpha(\eta)=1.
\]
The function \(\eta\) satisfies
\[
L\eta=-\lambda_0\eta,
\]
where \(L\) is the generator of the killed semigroup. The Q-process is then a Doob \(h\)-transform with \(h=\eta\) and exponential tilt \(e^{\lambda_0 t}\) [1404.1349].

At the level of path measures, the transformation is explicit:
\[
\left.\frac{dQ_x}{dP_x}\right|_{\mathcal F_s}
=
\mathbf{1}_{\{s<\tau_\partial\}}e^{\lambda_0 s}\frac{\eta(X_s)}{\eta(x)}.
\]
Its transition kernel is
\[
\tilde p(x;t,dy)=e^{\lambda_0 t}\frac{\eta(y)}{\eta(x)}p(x;t,dy),
\]
and the associated semigroup is
\[
\tilde P_t\varphi(x)=\frac{e^{\lambda_0 t}}{\eta(x)}P_t(\eta\varphi)(x).
\]
This formula makes precise that the Q-process is conservative even though the original dynamics are absorbed.

The invariant distribution of the Q-process is
\[
\beta(dx)=\eta(x)\alpha(dx),
\]
which needs no further normalization because \(\alpha(\eta)=1\). Moreover, the Q-process is exponentially ergodic:
\[
\left\|Q_{\mu_1}(X_t\in\cdot)-Q_{\mu_2}(X_t\in\cdot)\right\|_{TV}
\le (1-c_1c_2)^{\lfloor t/t_0\rfloor}\|\mu_1-\mu_2\|_{TV},
\]
and in particular
\[
\left\|Q_x(X_t\in\cdot)-\beta\right\|_{TV}
\le 2(1-c_1c_2)^{\lfloor t/t_0\rfloor}.
\]
The weak infinitesimal generator is correspondingly transformed:
\[
\tilde L^w f=\lambda_0 f+\frac{L^w(\eta f)}{\eta},
\]
on the domain
\[
{\cal D}(\tilde{L}^w)=
\left\{ f\in{\cal B}(E):\ \eta f\in{\cal D}(L^w)\text{ and }\frac{L^w(\eta f)}{\eta}\text{ is bounded} \right\}.
\]
A plausible implication is that the Q-process should be regarded not merely as a conditioned limit, but as the intrinsic conservative dynamics singled out by the principal survival eigenfunction.

## 4. Quantitative convergence to the Q-process

The asymptotic definition of the Q-process can be sharpened to an explicit finite-horizon approximation theorem. If there exist \(C,\gamma>0\) such that
\[
\left\|P_\mu(X_t\in\cdot\mid t<\tau_\partial)-\alpha\right\|_{TV}\le Ce^{-\gamma t},
\qquad \forall \mu,\ t\ge 0,
\]
then survival asymptotics satisfy the relative exponential estimate
\[
\left|e^{\lambda_0 t}P_x(t<\tau_\partial)-\eta(x)\right|
\le a_1\, e^{\lambda_0 t}P_x(t<\tau_\partial)e^{-\gamma t}
\]
for some \(a_1>0\). More importantly, for all \(t\ge 0\), all \(\Gamma\in\mathcal F_t\), and all \(T\ge t\),
\[
\left\|Q_x(\Gamma)-P_x(\Gamma\mid T<\tau_\partial)\right\|_{TV}\le a_2\, e^{-\gamma (T-t)}
\]
for some \(a_2>0\). Thus conditioning on survival up to a large but finite time \(T\) approximates the Q-process on any earlier time window with exponentially small error in the gap \(T-t\) [1611.02473].

This quantitative control yields a conditional ergodic theorem. If \(\beta=\eta\alpha\) is the invariant law of the Q-process and \(f\) is bounded measurable, then for the uniform measure on \([0,T]\),
\[
\left|E_x\left(\frac{1}{T}\int_0^T f(X_t)\,dt \mid T<\tau_\partial\right)-\int_E f\,d\beta\right|
\le \frac{a_4\,\|f\|_\infty}{T}.
\]
More generally, for any probability measure \(\mu_T\) on \([0,T]\),
\[
\left|E_x\left(\int_0^T f(X_t)\mu_T(dt)\mid T<\tau_\partial\right)-\int_E f\,d\beta\right|
\le a_3\|f\|_\infty \int_0^T\left(e^{-\gamma' t}+e^{-\gamma(T-t)}\right)\mu_T(dt),
\]
where \(\gamma'>0\) is the exponential ergodicity rate of the Q-process.

The converse direction is equally significant. If conditioned finite-time laws converge uniformly to a conservative Markov process \(Q\),
\[
\lim_{T\to\infty}\sup_{x\in E}
\left\|Q_x(X_t\in\cdot)-P_x(X_t\in \cdot\mid T<\tau_\partial)\right\|_{TV}=0,
\]
and this limit process is uniformly ergodic in the strong sense
\[
\lim_{t\to\infty}\sup_{x,y\in E}
\left\|Q_x(X_t\in\cdot)-Q_y(X_t\in\cdot)\right\|_{TV}=0,
\]
then the killed process admits a unique QSD and converges toward it exponentially fast, uniformly in its initial distribution [1611.02473]. This establishes an equivalence between exponentially mixing conditioned dynamics and the existence of a uniformly approximating ergodic Q-process.

## 5. Branching-process realizations and variants

For multitype Galton–Watson processes, the Q-process arises by conditioning on non-extinction in the remote future. If \((X_k)_{k\ge 0}\) is a \(d\)-type Galton–Watson process with mean matrix \(M\), Perron root \(p\), right eigenvector \(u\), and extinction time \(T\), then Nakagawa’s limit takes the form
\[
\lim_{n\to\infty} \mathbb P_{x_0}\!\left( X_{k_1}=x_1,\dots,X_{k_j}=x_j \mid X_{k_j+n}\neq 0,\ T<\infty \right)
=
\frac{x_j\cdot u}{x_0\cdot u}\, p^{k_j}\, \mathbb P_{x_0}(X_{k_1}=x_1,\dots,X_{k_j}=x_j).
\]
The limiting process is Markov with transition kernel
\[
Q_1(x,y)=\frac{y\cdot u}{x\cdot u}\,\frac{1}{p}\,P_1(x,y).
\]
In the positive recurrent regime its stationary measure is the size-biased Yaglom distribution
\[
\tilde \pi(z)=\frac{z\cdot u\,v(z)}{\sum_{y\neq 0} y\cdot u\,v(y)}.
\]
A substantial extension is that conditioning on reaching an accessible non-absorbing set \(S\), on reaching a positive threshold, or on hitting a nonzero state in the distant future yields the same Q-process. By contrast, conditioning on infinite total progeny leads to a process with the features of a Q-process, but it coincides with the original associated Q-process only in the critical regime [1412.3322].

A continuous-time branching analogue is developed for Markov branching population systems \(Z(t)\) on \(\mathbb N_0\), with extinction time \(H=\inf\{t:Z(t)=0\}\). The limiting transition probabilities
\[
L_{ij}(t)=\lim_{u\to\infty}\mathbf{P}_i\{Z(t+u)=j\mid u+t < H\}
\]
define a new Markov chain \(W(t)\), called the Markov Q-process. In this setting the structural parameter is
\[
B:=\exp\left\{-f'(q)\right\},
\]
where \(f\) is the infinitesimal generating function and \(q\) is the extinction probability of the original branching process. The Q-process transition generating function is
\[
G_i(t;x)=\mathbb{E}\!\left[x^{W(t)}\mid W(0)=i\right]
= \left(\Phi(t;qx)\right)^{i-1}\,e^{a_0 t}\,\frac{\Phi(t;x)}{q},
\]
and the local transition form is
\[
Q_{ij}(t)=\delta_{ij}+\pi_{ij}t+o(t),\qquad t\downarrow 0,
\]
with
\[
\pi_0=0,\qquad \pi_1=a_0-\ln B<0,\qquad \pi_j=j\,q^{j-1}a_j,\quad j\neq 1.
\]
The regimes are classified by \(B\): \(B=1\) gives a null recurrent Q-process, while \(B<1\) gives a positive recurrent one. The same paper studies structural-parameter estimation via the Lotka–Nagaev type estimator
\[
\widehat B(t) = \frac{W(t+1)-\mathbb{E}_1W(1)}{W(t)-1},
\]
which is exactly unbiased, and derives variance asymptotics in both regimes [2203.16904].

These branching realizations show that the Q-process is not tied to one conditioning scheme or one state space. The recurring mechanism is a harmonic tilt toward genealogies or trajectories that remain viable on long horizons.

## 6. Applications, scope, and terminological distinctions

The general criterion based on Assumption (A) has been verified in several model classes. For one-dimensional birth–death processes with catastrophes, the condition
\[
S:=\sum_{k\ge 1}\frac{1}{d_k\alpha_k}\sum_{l\ge k}\alpha_l<\infty,
\qquad
\alpha_k=\frac{\prod_{i=1}^{k-1}b_i}{\prod_{i=1}^{k}d_i},
\]
implies exponential convergence to a unique QSD, existence of the Q-process, and exponential ergodicity of the conditioned dynamics. The same abstract theory applies to multi-dimensional birth–death processes, infinite-dimensional population models with Brownian-type mutation, and neutron transport dynamics in a bounded domain, where the Q-process describes the motion conditioned on not yet being absorbed or having left the physical domain [1404.1349].

The term **Q-process** is, however, not universal across adjacent literatures. It is not synonymous with the **q-Hahn process**, which is an integrable stochastic interacting particle system with open boundaries and “no direct discussion of Q-processes” in the probabilistic sense [2205.10512]. It is also distinct from the **Q-Exponential Process**, a Bayesian prior on functions whose finite-dimensional marginals are consistent multivariate \(q\)-exponential laws [2210.07987], and from quantum-process tomography, which reconstructs unknown quantum channels from measurement data rather than conditioning absorbed stochastic dynamics on survival [1204.5936]. This suggests that “Q-process” has a precise technical meaning only within the quasi-stationary and long-time conditioning theory of absorbed Markov and branching systems.

In that probabilistic meaning, the Q-process is the canonical conservative dynamics extracted from an absorbed system by conditioning on eternal survival. Its mathematical content is the conjunction of a QSD, a principal survival eigenfunction, a Doob \(h\)-transform, and quantitative control of finite-horizon conditioning limits. Its principal significance is that it converts asymptotic survival behavior into an autonomous Markov process with its own invariant law and ergodic theory.

Source: https://www.emergentmind.com/topics/q-process