  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.
