Renormalization Analysis of interacting spatial stochastic systems
Greven A (2000)
Publication Type: Conference contribution, Original article
Publication year: 2000
Pages Range: 171-196
Conference Proceedings Title: Proceedings of the Royal Dutch Academy
Event location: Amsterdam
Abstract
We study large space-time scale properties of interacting spatial stochastic models arising in population genetics. Among those models are classical branching systems and resampling systems (interacting Feller's branching diffusions and interacting Fleming-Viot processes). We consider one-type, multi-type and 1 -type populations.
The tool of analysis is renormalization. This technique allows to describe large space-time scale properties of the stochastic system approximately by a discrete time stochastic process called the interaction chain. The transition kernels of this time-inhomogeneous Markov process are determined by the orbit of a certain nonlinear map acting on a function space, whose points specify the diffusion function. This structure allows to connect universal features in the stochastic system (i.e. features occurring in a large class of evolution mechanism) with those of the orbit of the iterations of this nonlinear map. Namely in many cases the orbits approach fixed points or after rescaling fixed shapes. These distinguished points correspond to well known explicitly solvable models such as interacting Fisher-Wright diffusions, Feller's branching diffusions or critical Ornstein-Uhlenbeck processes. In this way we can explain universality properties of the longtime behavior of a large class of models in population genetics and give a rigorous justification for the frequent use of special models since they reflect the typical qualitative behavior.
Finally we indicate how the renormalization analysis can be applied in evolutionary theory, i.e. models including migration, selection, mutation and recombination.
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How to cite
APA:
Greven, A. (2000). Renormalization Analysis of interacting spatial stochastic systems. In Ph. Clément, F. den Hollander, J. van Neerven, B. de Pagter (Eds.), Proceedings of the Royal Dutch Academy (pp. 171-196). Amsterdam, NL.
MLA:
Greven, Andreas. "Renormalization Analysis of interacting spatial stochastic systems." Proceedings of the Infinite-dimensional stochastic analysis, Amsterdam Ed. Ph. Clément, F. den Hollander, J. van Neerven, B. de Pagter, 2000. 171-196.
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