Error estimates on ergodic properties of discretized Feynman-Kac semigroups

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.