Tree based functional expansions for Feynman--Kac particle models

Pierre del Moral (1) , Frédéric Patras (2) , Sylvain Rubenthaler (2)

1 UNSW - University of New South Wales [Sydney]
2 JAD - Laboratoire Jean Alexandre Dieudonné

We design exact polynomial expansions of a class of Feynman--Kac particle distributions. These expansions are finite and are parametrized by coalescent trees and other related combinatorial quantities. The accuracy of the expansions at any order is related naturally to the number of coalescences of the trees. Our results include an extension of the Wick product formula to interacting particle systems. They also provide refined nonasymptotic propagation of chaos-type properties, as well as sharp $\mathbb{L}_p$-mean error bounds, and laws of large numbers for $U$-statistics.

Mots clés: combinatorial enumeration; tree; forest; Feynman-Kac semigroups; interacting particle systems

Domaines: Probabilités [math.PR]; Anneaux et algèbres [math.RA]

Pierre del Moral, Frédéric Patras, Sylvain Rubenthaler. Tree based functional expansions for Feynman--Kac particle models. The Annals of Applied Probability, 2009, pp.778-825. ⟨10.1214/08-AAP565⟩. ⟨hal-00086532⟩