Error estimates on ergodic properties of discretized Feynman-Kac semigroups

TL;DR

This paper provides error estimates for the discretization of Feynman-Kac semigroups related to diffusion processes, aiding the development of efficient numerical schemes and validating previous empirical findings.

Contribution

It introduces error bounds for invariant measures and eigenvalues of discretized Feynman-Kac semigroups, offering theoretical support for numerical methods in various applications.

Findings

Error estimates for invariant measures of discretized semigroups

Error bounds for the leading eigenvalue of the generator

Numerical simulations confirming theoretical results

Abstract

We consider the numerical analysis of the time discretization of Feynman-Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present errors estimates a la Talay-Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman-Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.