Genealogies and Increasing Propagation of Chaos For Feynman-Kac and Genetic Models

A path-valued interacting particle systems model for the genealogical structure of genetic algorithms is presented. We connect the historical process and the distribution of the whole ancestral tree with a class of Feynman-Kac formulae on path space. We also prove increasing and uniformversions of propagation of chaos for appropriate particle block size and time horizon yielding what seems to be the first result of this type for this class of particle systems. 1. Introduction. Over the last two decades themhave been im portant developments centering around the connections between genetic algorithms and Feynman-Kac formulae. This subject has natural links to biology, evolutionary computing, physics and advanced signal processing. The reader who wishes to know more details about these connections and specific applications is recommended to consult the survey paper [8] and references therein. In the previously referenced paper we essentially discussed the asymptotic behavior of the empirical measures associated to genetic-type particle systems as the number of particles tends to infinity. The strong versions of propagation of chaos presented here provide several measures of centrality and asymptotic independence for the distribution of a block of particles up to a given time horizon. These asymptotic results complement and strengthen those presented in [8]. Another side topic of the present work concerns the modeling and the convergence analysis of the historical process in population genetics. Aside from inherent and mathematical interest one of the practical reasons for studying the genealogical structure of a genetic algorithmstems fromthe fact that this set up is precisely what we need to solve numerically the so-called non linear filtering and smoothing problem. This opening section is decomposed into three parts. We begin in Section 1.1 with the Feynman-Kac formulae and provide a brief description of the corresponding genetic-type interacting particle system approximating model. In Section 1.2 we describe in some details the main results of the paper. In Section 1.3 we close with some comments on related works on the subject and some open problems. Here are some standard notations to be used in all the paper. Let � � E� �

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