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

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 $\forall t\in \mathbf {N} ,\operatorname {P} _{\nu }(T>t)=\exp(-\theta (\nu )\times t)$.

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.