This paper focuses on interacting particle systems methods for solving numerically a class of Feynman-Kac formulae arising in the study of certain parabolic differential equations, physics, biology, evolutionary computing, nonlinear filtering and elsewhere. We have tried to give an “expose” of the mathematical theory that is useful for analyzing the convergence of such genetic-type and particle approximating models including law of large numbers, large deviations principles, fluctuations and empirical process theory as well as semigroup techniques and limit theorems for processes.

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Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering

The stability properties of a class of interacting measure valued processes arising in nonlinear filtering and genetic algorithm theory is discussed. Simple sufficient conditions are given for exponential decays. These criteria are applied to study the asymptotic stability of the nonlinear filtering equation and infinite population models as those arising in Biology and evolutionary computing literature. On the basis of these stability properties we also propose a uniform convergence theorem for the interacting particle numerical scheme of the nonlinear filtering equation introduced in a previous work. In the last part of this study we propose a refinement genetic type particle method with periodic selection dates and we improve the previous uniform convergence results. We finally discuss the uniform convergence of particle approximations including branching and random population size systems.

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On the stability of interacting processes with applications to filtering and genetic algorithms

Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds

This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject. Table of Contents: Background material; Sobolev spaces on $\mathbb{R}^n$; Differentiable measures on linear spaces; Some classes of differentiable measures; Subspaces of differentiability of measures; Integration by parts and logarithmic derivatives; Logarithmic gradients; Sobolev classes on infinite dimensional spaces; The Malliavin calculus; Infinite dimensional transformations; Measures on manifolds; Applications; References; Subject index. (Surv/164)

Differentiable Measures and the Malliavin Calculus

We study the stability of the optimal filter w.r.t. its initial condition and w.r.t. the model for the hidden state and the observations in a general hidden Markov model, using the Hilbert projective metric. These stability results are then used to prove, under some mixing assumption, the uniform convergence to the optimal filter of several particle filters, such as the interacting particle filter and some other original particle filters.

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Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters

Renormalized Differential Geometry on Path Space: Structural Equation, Curvature

Radonification of the heat equation associated to the Gelfand-Fuks cocycle is realized in a space of Holderian homeomorphism.

P. Malliavin

Papers

Renormalized Differential Geometry on Path Space: Structural Equation, Curvature

The canonic diffusion above the diffeomorphism group of the circle

Compact families of Wiener functionals

Analytic Functions, Cauchy Formula, and Stationary Phase on a Real Abstract Wiener Space

Non perturbative construction of invariant measure through confinement by curvature