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

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

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

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 $\nu$ is a QSD then there exists $\theta (\nu )>0$ such that $\mathrm{\forall }t\in 𝐍,{\mathrm{P}}_{\nu }(T>t)=\exp (-\theta (\nu )\times t)$.

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

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

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

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