Feynman–Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrödinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrödinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability analysis 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. We show that these conditions are satisfied in a variety of situations, including stochastic differential equations. 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.

Feynman-Kac半群出现在数学的各个领域：非线性滤波，大偏差理论，薛定ding算子的频谱分析等。它们的长时间行为提供了重要的信息，例如在Schrödinger算子的基态能量方面，或者在大偏差理论中按比例缩放了累积量生成函数。在本文中，我们为这些非线性演化提出了马尔可夫链稳定性分析的简单自然的扩展。与其他经典的遍历性结果一样，它基于两个假设：一个Lyapunov条件引起某种紧缩性，以及一个简化条件确保某些混合性。我们证明在各种情况下都满足这些条件，包括随机微分方程。提供了说明性示例，非线性半群的稳定性来自基础动力学或费曼-卡克权函数。我们还使用我们的技术为随机微分方程的离散化提供统一的时间步收敛估计。