A perturbation theory for Feynman-Kac models and their particle interpretations
Abstract
We develop a perturbation analysis of Feynman--Kac models, describing how local errors introduced at each step of the Feynman--Kac recursion propagate over time. Under general conditions on the potential functions and the Markov transitions, which are shown to hold for a large class of state-space models, we prove that the global error for any time horizon T≥ 1 is bounded by C ΔTlog(1+ΔT−1), where ΔT is the largest local error; this bound holds both for the prediction and the updated models. These conditions require no Lyapunov function and no stability of the Markov transitions, and allow for non-compact state spaces: confinement is provided by the potential functions. Our perturbation bounds hold in total variation and in the Kolmogorov norm; for the latter, we extend the Hardy--Krause variation and the Koksma--Hlawka inequality to unbounded domains. Our perturbation theory is applied to derive time-uniform bounds for linearized Kalman filters, Euler schemes for partially observed diffusions and nudged state-space models, as well as high-probability error bounds, uniform over whole trajectories, for particle filters and, more generally, for mean field particle methods such as sequential Monte Carlo and diffusion Monte Carlo algorithms.
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BibTeX
@misc{moral2026perturbation,
title = {{A perturbation theory for Feynman-Kac models and their particle interpretations}},
author = {Pierre Del Moral and Mathieu Gerber},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21587},
eprint = {2610.21587},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21587},
}
RIS
TY - UNPB
TI - A perturbation theory for Feynman-Kac models and their particle interpretations
AU - Pierre Del Moral
AU - Mathieu Gerber
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21587
UR - https://arxiv.org/abs/2610.21587
AB - We develop a perturbation analysis of Feynman--Kac models, describing how local errors introduced at each step of the Feynman--Kac recursion propagate over time. Under general conditions on the potential functions and the Markov transitions, which are shown to hold for a large class of state-space models, we prove that the global error for any time horizon T≥ 1 is bounded by C Δ_Tlog(1+Δ_T^−1), where Δ_T is the largest local error; this bound holds both for the prediction and the updated models. These conditions require no Lyapunov function and no stability of the Markov transitions, and allow for non-compact state spaces: confinement is provided by the potential functions. Our perturbation bounds hold in total variation and in the Kolmogorov norm; for the latter, we extend the Hardy--Krause variation and the Koksma--Hlawka inequality to unbounded domains. Our perturbation theory is applied to derive time-uniform bounds for linearized Kalman filters, Euler schemes for partially observed diffusions and nudged state-space models, as well as high-probability error bounds, uniform over whole trajectories, for particle filters and, more generally, for mean field particle methods such as sequential Monte Carlo and diffusion Monte Carlo algorithms.
ER -