Tree based functional expansions for Feynman--Kac particle models

TL;DR

This paper introduces exact polynomial expansions for Feynman--Kac particle distributions using coalescent trees, extending Wick formulas and providing precise error bounds and propagation properties.

Contribution

It develops finite, exact polynomial expansions parametrized by coalescent trees for Feynman--Kac particle models, extending Wick formulas and improving error analysis.

Findings

Finite polynomial expansions parametrized by coalescent trees.

Extension of Wick product formula to interacting particle systems.

Sharp error bounds and propagation of chaos results.

Abstract

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.