Course 1: Quasi-Stationary Distributions for absorbed
Markov Processes
Markov
processes almost surely absorbed in finite time arise in many areas of
applications, particularly populations dynamics/genetics where absorption
corresponds
to the extinction of a (sub)population. Such processes often exhibit a
stabilization of the population distribution before absorption
(e.g. the mortality plateau in living populations), although
their stationary distribution is trivial.
To study
this phenomenon, the proper notion is the one of quasi-stationary
distributions, i.e. distributions stationary conditionally on non-absorption.
The aim of
this course is to present the basic properties of quasi-stationary
distributions and to study the properties of existence, uniqueness,
and
convergence of conditional distributions for several classes of classical
processes in population dynamics (birth and death processes, Wright-Fisher
diffusions of
Fisher diffusions with interaction... in one and several dimensions). We will in particular present recent results making
use of probabilistic coupling techniques developed
in
collaboration with Denis Villemonais. The question of
approximation of quasi-stationary distributions with particle systems will also
be studied.
