Abstract
Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. This article presents a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula allows us to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein [2] in the context of discrete generation models to continuous time Feynman-Kac models and their interacting jump particle interpretations. We also provide new propagation of chaos estimates for continuous time genealogical tree based particle models with respect to the time horizon and the size of the systems. These results allow to obtain sharp quantitative estimates of the convergence rate to equilibrium of particle Gibbs-Glauber samplers. To the best of our knowledge these results are the first of this kind for continuous time Feynman-Kac measures.
Highlights
Feynman-Kac measures on path spaces are central in applied probability as well as in biology and quantum physics
For any given x P DtpSq and z “ pzsqsďt P DtpSp SpN1q, we summarize the transition of the particle Gibbs sampler graphically as follows:
We provide a detailed discussion on some numerical aspects of the particle Gibbs-Glauber dynamics introduced above as well as some comparisons with existing literature on interacting particle systems
Summary
Feynman-Kac measures on path spaces are central in applied probability as well as in biology and quantum physics. Their discrete time versions are encapsulate a variety of well known algorithms such as particle filters [23] (a.k.a. sequential Monte Carlo methods in Bayesian literature [14, 21, 22, 33, 40]), the go-with the winner [1], as well as the self-avoidind random walk pruned-enrichment algorithm by Rosenbluth and Rosenbluth [69], and many others. These techniques provide with little efforts new uniform propagation of chaos estimates w.r.t. the time horizon (cf. corollary 3.13)
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