Coalescent tree based functional representations for some Feynman-Kac particle models

We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp $\LL\_p$-mean error bounds, and laws of large numbers for $U$-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schr\"{o}dinger semigroups are also discussed.