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

The general definition^[1] is: a probability measure ${\displaystyle \nu }$ on ${\displaystyle {\mathcal {X}}^{a}}$ is said to be a quasi-stationary distribution (QSD) if for every measurable set ${\displaystyle B}$ contained in ${\displaystyle {\mathcal {X}}^{a}}$, $${\displaystyle \forall t\geq 0,\operatorname {P} _{\nu }(Y_{t}\in B\mid T>t)=\nu (B)}$$where ${\displaystyle \operatorname {P} _{\nu }=\int _{{\mathcal {X}}^{a}}\operatorname {P} _{x}\,\mathrm {d} \nu (x)}$.

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

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:^[1][2] if ${\displaystyle \nu }$ is a QSD then there exists ${\displaystyle \theta (\nu )>0}$ such that ${\displaystyle \forall t\in \mathbf {N} ,\operatorname {P} _{\nu }(T>t)=\exp(-\theta (\nu )\times t)}$.

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

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

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

A sufficient condition for a QSD to exist is given considering the transition semigroup ${\displaystyle (P_{t},t\geq 0)}$ of the process before killing. Then, under the conditions that ${\displaystyle {\mathcal {X}}^{a}}$ is a compact Hausdorff space and that ${\displaystyle P_{1}}$ preserves the set of continuous functions, i.e. ${\displaystyle P_{1}({\mathcal {C}}({\mathcal {X}}^{a}))\subseteq {\mathcal {C}}({\mathcal {X}}^{a})}$, there exists a QSD.

History

The works of Wright on gene frequency in 1931^[3] and of Yaglom on branching processes in 1947^[4] already included the idea of such distributions. The term quasi-stationarity applied to biological systems was then used by Bartlett in 1957,^[5] who later coined "quasi-stationary distribution".^[6]

Quasi-stationary distributions were also part of the classification of killed processes given by Vere-Jones in 1962^[7] and their definition for finite state Markov chains was done in 1965 by Darroch and Seneta.^[8]

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.