Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
Other Grants and Activities

Section: New Results

Mean field interacting particle systems

Participants : Ajay Jasra, Frédéric Patras, Pierre Del Moral, Arnaud Doucet, Sylvain Rubenthaler.

The design and the mathematical analysis of genetic type and branching particle interpretations of Feynman- Kac-Schroedinger type semigroups (and vice versa) has been developed by group of researchers since the beginning of the 90's. In Bayesian statistics these sampling technology is also called sequential Monte Carlo methods. For further details, we refer to the books  [33] , [37] , [26] , and references therein. This Feynman-Kac particle methodology is increasingly identified with emerging subjects of physics, biology, and engineering science. This new theory on genetic type branching and interacting particle systems has led to spectacular results in signal processing and in quantum chemistry with precise estimates of the top eigenvalues, and the ground states of Schrodinger operators. It offers a rigorous and unifying mathematical framework to analyze the convergence of a variety of heuristic-like algorithms currently used in biology, physics and engineering literature since the beginning of the 1950's. It applies to any stochastic engineering problem which can be translated in terms of functional Feynman-Kac type measures. During the last two decades the range of application of this modern approach to Feynman-Kac models has increased revealing unexpected applications in a number of scientific disciplines including in : The analysis of Dirichlet problems with boundary conditions, financial mathematics, molecular analysis, rare events and directed polymers simulation, genetic algorithms, Metropolis-Hastings type models, as well as filtering problems and hidden Markov chains.

In the period 2008-2009, these lines of research have been developed in three different directions. To develop a concrete peer to peer interaction between applied mathematics, engineering and computer sciences, we have written two review and pedagogical book chapters on stochastic particle algorithms and their applications ([12] , [17] ). Our second line of research is concerned with the foundations and the mathematical analysis of mean field particle models. In 2009, we have developed a series of new important results such as exact propagation of chaos expansions of the law of particle blocks ([7] , [21] ), sharp and non asymptotic exponential concentration inequalities [22] , and the refined stability analysis of neutral type genetic models [6] . The third line of our research is concerned with the design of new classes of stochastic particle algorithms, including particle approximate Bayesian computation [18] , backward Feynman-Kac particle models [19] , particle approximations of fixed parameters in hidden Markov chain models [20] , and particle rare event simulation of static probability laws [16] . The details of these contributions are provided below.

Review book chapters

Stochastic analysis

New class of particle algorithms


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