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

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. (C) 2001 Editions scientifiques et medicales Elsevier SAS.