Sharp propagation of chaos estimates for Feynmann–Kac particle models

This article is concerned with the propagation-of-chaos properties of genetic-type particle models. This class of models arises in a variety of scientific disciplines including theoretical physics, macromolecular biology, engineering sciences, and more particularly in computational statistics and advanced signal processing. From the pure mathematical point of view, these interacting particle systems can be regarded as a mean field particle interpretation of a class of Feynman-Kac measures on path spaces. In the present paper, we design an original integration theory of propagation of chaos based on the fluctuation analysis of a class of interacting particle random fields. We provide analytic functional representations of the distributions of finite particle blocks, yielding what seems to be the first result of this kind for interacting particle systems. These asymptotic expansions are expressed in terms of the limiting Feynman-Kac semigroups and a class of interacting jump operators. These results provide both sharp estimates of the negligible bias introduced by the interaction mechanisms, and central limit theorems for nondegenerate $U$-statistics and von Mises statistics associated with genealogical tree models. Applications to nonlinear filtering problems and interacting Markov chain Monte Carlo algorithms are discussed.