Q-Process in Markov and Branching Systems

• Q-Process is a conditioned Markov process obtained by reweighting path probabilities to model eternal survival in absorbed systems using quasi-stationary distributions.
• It employs a Doob h-transform and uniform exponential convergence estimates to establish conservative dynamics and quantify survival asymptotics.
• The framework extends to branching processes by conditioning on non-extinction, providing precise ergodic properties and convergence rates.

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 (Champagnat et al., 2014, Champagnat et al., 2016). 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 (Pénisson, 2014, Imomov et al., 2022).

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 $E\cup\{\partial\}$0 almost surely under every $E\cup\{\partial\}$1, while $E\cup\{\partial\}$2 for all $E\cup\{\partial\}$3 and $E\cup\{\partial\}$4. A probability measure $E\cup\{\partial\}$5 on $E\cup\{\partial\}$6 is a quasi-stationary distribution if

$E\cup\{\partial\}$7

If $E\cup\{\partial\}$8, the conditioned law is therefore invariant at every time. When $E\cup\{\partial\}$9 is a QSD, there exists $\partial$0 such that

$\partial$1

This exponential survival law identifies the decay rate governing the absorbed process and simultaneously supplies the spectral parameter entering the Q-process construction (Champagnat et al., 2014).

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 $\partial$2 on $\partial$3 such that

$\partial$4

for some $\partial$5 and all $\partial$6, and

$\partial$7

for some $\partial$8, all $\partial$9, and all $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$0. The first condition is a uniform minorization for the conditioned process; the second requires survival from $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$1 to be uniformly comparable to survival from any point. Equivalent variants, denoted (A′) and (A′′), are also available (Champagnat et al., 2014).

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 $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$2 and constants $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$3 such that

$\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$4

the corresponding pointwise form for all initial states $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$5; and the integrability condition

$\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$6

Whenever these equivalent properties hold, $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$7 is the unique QSD, convergence is exponential and uniform in the initial law, and one obtains the explicit estimate

$\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$8

The same hypotheses also ensure existence and exponential ergodicity of the Q-process itself (Champagnat et al., 2014).

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 $\tau_\partial:=\inf\{t\ge 0:X_t=\partial\}$9-transform structure of the Q-process

Under the preceding hypotheses, survival probabilities admit a uniform asymptotic renormalization through a positive eigenfunction

$$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$0

The function $$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$1 satisfies

$$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$2

where $$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$3 is the generator of the killed semigroup. The Q-process is then a Doob $$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$4-transform with $$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$5 and exponential tilt $$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$6 (Champagnat et al., 2014).

At the level of path measures, the transformation is explicit: $$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$7 Its transition kernel is

$$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$8

and the associated semigroup is

$$Q_x(A)=\lim_{t\to\infty}P_x(A\mid t<\tau_\partial),\qquad A\in\mathcal F_s,\ s\ge 0.$$9

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

$h$0

which needs no further normalization because $h$1. Moreover, the Q-process is exponentially ergodic: $h$2 and in particular

$h$3

The weak infinitesimal generator is correspondingly transformed: $h$4 on the domain

$h$5

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 $h$6 such that

$h$7

then survival asymptotics satisfy the relative exponential estimate

$h$8

for some $h$9. More importantly, for all $(X_t)_{t\ge 0}$0, all $(X_t)_{t\ge 0}$1, and all $(X_t)_{t\ge 0}$2,

$(X_t)_{t\ge 0}$3

for some $(X_t)_{t\ge 0}$4. Thus conditioning on survival up to a large but finite time $(X_t)_{t\ge 0}$5 approximates the Q-process on any earlier time window with exponentially small error in the gap $(X_t)_{t\ge 0}$6 (Champagnat et al., 2016).

This quantitative control yields a conditional ergodic theorem. If $(X_t)_{t\ge 0}$7 is the invariant law of the Q-process and $(X_t)_{t\ge 0}$8 is bounded measurable, then for the uniform measure on $(X_t)_{t\ge 0}$9,

$E\cup\{\partial\}$0

More generally, for any probability measure $E\cup\{\partial\}$1 on $E\cup\{\partial\}$2,

$E\cup\{\partial\}$3

where $E\cup\{\partial\}$4 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 $E\cup\{\partial\}$5,

$E\cup\{\partial\}$6

and this limit process is uniformly ergodic in the strong sense

$E\cup\{\partial\}$7

then the killed process admits a unique QSD and converges toward it exponentially fast, uniformly in its initial distribution (Champagnat et al., 2016). 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 $E\cup\{\partial\}$8 is a $E\cup\{\partial\}$9-type Galton–Watson process with mean matrix $\partial$0, Perron root $\partial$1, right eigenvector $\partial$2, and extinction time $\partial$3, then Nakagawa’s limit takes the form

$\partial$4

The limiting process is Markov with transition kernel

$\partial$5

In the positive recurrent regime its stationary measure is the size-biased Yaglom distribution

$\partial$6

A substantial extension is that conditioning on reaching an accessible non-absorbing set $\partial$7, 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 (Pénisson, 2014).

A continuous-time branching analogue is developed for Markov branching population systems $\partial$8 on $\partial$9, with extinction time $$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$0. The limiting transition probabilities

$$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$1

define a new Markov chain $$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$2, called the Markov Q-process. In this setting the structural parameter is

$$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$3

where $$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$4 is the infinitesimal generating function and $$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$5 is the extinction probability of the original branching process. The Q-process transition generating function is

$$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$6

and the local transition form is

$$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$7

with

$$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$8

The regimes are classified by $$X_s=\partial \implies X_t=\partial \quad \forall t\ge s.$$9: $E\cup\{\partial\}$00 gives a null recurrent Q-process, while $E\cup\{\partial\}$01 gives a positive recurrent one. The same paper studies structural-parameter estimation via the Lotka–Nagaev type estimator

$E\cup\{\partial\}$02

which is exactly unbiased, and derives variance asymptotics in both regimes (Imomov et al., 2022).

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

$E\cup\{\partial\}$03

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 (Champagnat et al., 2014).

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 (Frassek, 2022). It is also distinct from the Q-Exponential Process, a Bayesian prior on functions whose finite-dimensional marginals are consistent multivariate $E\cup\{\partial\}$04-exponential laws (Li et al., 2022), and from quantum-process tomography, which reconstructs unknown quantum channels from measurement data rather than conditioning absorbed stochastic dynamics on survival (Anis et al., 2012). 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 $E\cup\{\partial\}$05-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.