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

TL;DR

This paper introduces a novel tree-based functional framework for Feynman-Kac particle models, extending Wick formulas and providing refined error bounds and laws of large numbers, with applications to quantum mechanics.

Contribution

It develops a new combinatorial, permutation group-based representation for particle distributions, extending Wick formulas and improving error and convergence analysis.

Findings

Refined non-asymptotic propagation of chaos results

Sharp $ ext{L}_p$-mean error bounds for particle systems

Laws of large numbers for U-statistics in the context of Feynman-Kac models

Abstract

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.