Quasi-stationary distribution

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A quasi-stationary distribution (QSD) describes the long-run behavior of a Markov process that has absorbing states (states that, once entered, cannot be left). The idea is that, although the process will eventually be absorbed, its distribution can look stable for a long time if it starts in the right way.

A common example is a population that will eventually go extinct, but the number of individuals can stay roughly constant for a long time before collapse.

Setup in simple terms:
- We have a Markov process Y_t with possible states X. Some states are absorbing (once reached, the process stays there) and the rest are non-absorbing.
- Let T be the first time the process hits an absorbing state (the “killing” time). For any starting state x, P_x denotes the probability law of the process.
- It is assumed that absorption happens with probability 1, i.e., T is finite almost surely.

What is a QSD?
- A QSD is a probability distribution ν on the non-absorbing states such that, if you start the process with Y_0 distributed according to ν and condition on not yet being absorbed by time t, the distribution of Y_t remains ν for every t ≥ 0.
- In symbols: for every measurable set B of non-absorbing states and every t ≥ 0, Pν(Y_t ∈ B | T > t) = ν(B). Here Pν is the law obtained by starting with ν and letting the process run.

Consequences of a QSD:
- The distribution of the process, given that it has not been absorbed yet, does not change over time.
- The time to absorption, under a QSD, is typically exponential: there is a rate θ(ν) > 0 such that Pν(T > t) = exp(-θ(ν) t). For any smaller rate, certain moment conditions still hold.
- A common question is when a QSD exists for a given system.

Existence conditions (at a glance):
- Define θx* = sup{ θ : E_x[e^{θ T}] < ∞ } for a non-absorbing starting state x. A necessary condition for a QSD to exist is that θx* > 0 for some x, and θx* = liminf as t→∞ of -(1/t) log P_x(T > t).
- If a QSD ν exists, then Eν[e^{θ(ν) T}] = ∞.
- There are also sufficient conditions. For example, if the non-absorbing set is compact and the Markov semigroup P_t preserves continuity (so P1 maps continuous functions to continuous functions), then a QSD exists.

Historical notes:
- The idea goes back to Wright (gene frequency) and Yaglom (branching processes). The term “quasi-stationary distribution” was used by Bartlett in 1957. Quasi-stationary distributions were also discussed in the classification of killed processes by Vere-Jones and formalized for finite-state Markov chains by Darroch and Seneta.

Applications:
- QSDs are used to model processes that can persist for a long time before eventually dying out, such as population dynamics, disease spread, chemical reactions, and other systems with absorbing states. They capture the meaningful “long-lived” behavior before absorption.