More on the long time stability of Feynman-Kac semigroups

TL;DR

This paper extends classical stability results to Feynman-Kac semigroups, providing conditions for their long-term stability and applying these to discretized stochastic differential equations.

Contribution

It introduces a natural extension of Markov chain stability to non-linear Feynman-Kac semigroups using Lyapunov and minorization conditions.

Findings

Established stability criteria for Feynman-Kac semigroups.

Provided examples demonstrating stability from dynamics or weights.

Achieved uniform convergence estimates for discretized SDEs.

Abstract

Feynman-Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compactness, and a minorization condition ensuring some mixing. Illustrative examples are provided, where the stability of the non-linear semigroup arises either from the underlying dynamics or from the Feynman-Kac weight function. We also use our technique to provide uniform in the time step convergence estimates for discretizations of stochastic differential equations