Quasi-stationary distribution

Short description: Type of random process

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

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

In particular [math]\displaystyle{ \forall B \in \mathcal{B}(\mathcal{X}^a), \forall t \geq 0, \operatorname{P}_\nu(Y_t \in B, T \gt t) = \nu(B) \operatorname{P}_\nu(T \gt t). }[/math]

General results

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 [math]\displaystyle{ \nu }[/math] is a QSD then there exists [math]\displaystyle{ \theta(\nu) \gt 0 }[/math] such that [math]\displaystyle{ \forall t \in \mathbf{N}, \operatorname{P}_\nu(T \gt t) = \exp(-\theta(\nu) \times t) }[/math].

Moreover, for any [math]\displaystyle{ \vartheta \lt \theta(\nu) }[/math] we get [math]\displaystyle{ \operatorname{E}_\nu(e^{\vartheta t}) \lt \infty }[/math].

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

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

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