A Backward Particle Interpretation of Feynman-Kac Formulae

TL;DR

This paper introduces a novel backward particle interpretation of Feynman-Kac measures that enables efficient computation of additive functionals and occupation measures over long time horizons, with proven convergence and concentration results.

Contribution

It proposes a new backward Markovian particle algorithm that improves upon traditional genealogical models for Feynman-Kac measures, allowing on-the-fly computations and uniform convergence.

Findings

Uniform convergence with respect to time horizon

Functional central limit theorems established

Exponential concentration estimates proven

Abstract

We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals "on-the-fly" as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates. We also illustrate these results in the context of computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to $h$-processes.