Dawson DA, Greven A, Vaillancourt J (1995)
Publication Language: English
Publication Type: Journal article, Original article
Publication year: 1995
Publisher: American Mathematical Society
Book Volume: 347
Pages Range: 2277-2361
Journal Issue: 7
URI: http://www.ams.org/journals/tran/1995-347-07/S0002-9947-1995-1297523-5/
DOI: 10.1090/S0002-9947-1995-1297523-5
In this paper of infinite systems of interacting measure-valued dif-
fusions each with state space ¿^([O, 1]), the set of probability measures on
[0, 1], is constructed and analysed (Fleming-Viot systems). These systems arise
as diffusion limits of population genetics models with infinitely many possible
types of individuals (labelled by [0, 1]), spatially distributed over a countable
collection of sites and evolving as follows. Individuals can migrate between
sites and after an exponential waiting time a colony replaces its population by a
new generation where the types are assigned by resampling from the empirical
distribution of types at this site.
It is proved that, depending on recurrence versus transience properties of the
migration mechanism, the system either clusters as r —> oo , that is, converges
in distribution to a law concentrated on the states in which all components are
equal to some Su , « £ [0, 1], or the system approaches a nontrivial equi-
librium state. The properties of the equilibrium states, respectively the cluster
formation, are studied by letting a parameter in the migration mechanism tend
to infinity and explicitly identifying the limiting dynamics in a sequence of dif-
ferent space-time scales. These limiting dynamics have stationary states which
are quasi-equiiibria of the original system, that is, change only in longer time
scales. Properties of these quasi-equilibria are derived and related to the global
equilibrium process for large N. Finally we establish that the Fleming-Viot
systems are the unique dynamics which remain invariant under the associated
space-time renormalization procedure.
APA:
Dawson, D.A., Greven, A., & Vaillancourt, J. (1995). Equilibria and quasi-equlibria for infinite systems of Feming-Viot processes. Transactions of the American Mathematical Society, 347(7), 2277-2361. https://dx.doi.org/10.1090/S0002-9947-1995-1297523-5
MLA:
Dawson, Donald Andrew, Andreas Greven, and Jean Vaillancourt. "Equilibria and quasi-equlibria for infinite systems of Feming-Viot processes." Transactions of the American Mathematical Society 347.7 (1995): 2277-2361.
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