Quasi-stationary distribution

In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses.

Formal definition
We consider a Markov process $$(Y_t)_{t \geq 0}$$ taking values in $$\mathcal{X}$$. There is a measurable set $$\mathcal{X}^{\mathrm{tr}}$$of absorbing states and $$\mathcal{X}^a = \mathcal{X} \setminus \mathcal{X}^{\operatorname{tr}}$$. We denote by $$T$$ the hitting time of $$\mathcal{X}^{\operatorname{tr}}$$, also called killing time. We denote by $$\{ \operatorname{P}_x \mid x \in \mathcal{X} \}$$ the family of distributions where $$\operatorname{P}_x$$ has original condition $$Y_0 = x \in \mathcal{X}$$. We assume that $$\mathcal{X}^{\operatorname{tr}}$$ is almost surely reached, i.e. $$\forall x \in \mathcal{X}, \operatorname{P}_x(T < \infty) = 1$$.

The general definition is: a probability measure $$\nu$$ on $$\mathcal{X}^a$$ is said to be a quasi-stationary distribution (QSD) if for every measurable set $$B$$ contained in $$\mathcal{X}^a$$, $$\forall t \geq 0, \operatorname{P}_\nu(Y_t \in B \mid T > t) = \nu(B)$$where $$\operatorname{P}_\nu = \int_{\mathcal{X}^a} \operatorname{P}_x \, \mathrm{d} \nu(x)$$.

In particular $$\forall B \in \mathcal{B}(\mathcal{X}^a), \forall t \geq 0, \operatorname{P}_\nu(Y_t \in B, T > t) = \nu(B) \operatorname{P}_\nu(T > t).$$

Killing time
From the assumptions above we know that the killing time is finite with probability 1. A stronger result than we can derive is that the killing time is exponentially distributed: if $$\nu$$ is a QSD then there exists $$\theta(\nu) > 0$$ such that $$\forall t \in \mathbf{N}, \operatorname{P}_\nu(T > t) = \exp(-\theta(\nu) \times t)$$.

Moreover, for any $$\vartheta < \theta(\nu)$$ we get $$\operatorname{E}_\nu(e^{\vartheta t}) < \infty$$.

Existence of a quasi-stationary distribution
Most of the time the question asked is whether a QSD exists or not in a given framework. From the previous results we can derive a condition necessary to this existence.

Let $$\theta_x^* := \sup \{ \theta \mid \operatorname{E}_x(e^{\theta T}) < \infty \}$$. A necessary condition for the existence of a QSD is $$\exists x \in \mathcal{X}^a, \theta_x^* > 0$$ and we have the equality $$\theta_x^* = \liminf_{t \to \infty} -\frac{1}{t} \log(\operatorname{P}_x(T > t)).$$

Moreover, from the previous paragraph, if $$\nu$$ is a QSD then $$\operatorname{E}_\nu \left( e^{\theta(\nu)T} \right) = \infty$$. As a consequence, if $$\vartheta > 0$$ satisfies $$\sup_{x \in \mathcal{X}^a} \{ \operatorname{E}_x(e^{\vartheta T}) \} < \infty$$ then there can be no QSD $$\nu$$ such that $$\vartheta = \theta(\nu)$$ because other wise this would lead to the contradiction $$\infty = \operatorname{E}_\nu \left( e^{\theta(\nu)T} \right) \leq \sup_{x \in \mathcal{X}^a} \{ \operatorname{E}_x(e^{\theta(\nu) T}) \} < \infty $$.

A sufficient condition for a QSD to exist is given considering the transition semigroup $$(P_t, t \geq 0)$$ of the process before killing. Then, under the conditions that $$\mathcal{X}^a$$ is a compact Hausdorff space and that $$P_1$$ preserves the set of continuous functions, i.e. $$P_1(\mathcal{C}(\mathcal{X}^a)) \subseteq \mathcal{C}(\mathcal{X}^a)$$, there exists a QSD.

History
The works of Wright on gene frequency in 1931 and of Yaglom on branching processes in 1947 already included the idea of such distributions. The term quasi-stationarity applied to biological systems was then used by Bartlett in 1957, who later coined "quasi-stationary distribution".

Quasi-stationary distributions were also part of the classification of killed processes given by Vere-Jones in 1962 and their definition for finite state Markov chains was done in 1965 by Darroch and Seneta.

Examples
Quasi-stationary distributions can be used to model the following processes:
 * Evolution of a population by the number of people: the only equilibrium is when there is no one left.
 * Evolution of a contagious disease in a population by the number of people ill: the only equilibrium is when the disease disappears.
 * Transmission of a gene: in case of several competing alleles we measure the number of people who have one and the absorbing state is when everybody has the same.
 * Voter model: where everyone influences a small set of neighbors and opinions propagate, we study how many people vote for a particular party and an equilibrium is reached only when the party has no voter, or the whole population voting for it.