通过Föllmer漂移的1-Wasserstein距离中的定量QSD收敛

中文摘要 AI 辅助

我们发展了一种新颖的路径wise方法来研究被杀死扩散过程在非吸收条件下的条件分布随时间趋于无穷时向拟平稳分布（QSD）的收敛。我们从一般观察出发：一个被吸收的马尔可夫过程在存活至时间T>0条件下的动力学，是在该时间点一个简单分布约束下，相对于其无条件动力学的路径wise相对熵的最小化者；换言之，即一个Föllmer过程。然后我们展示这一结果如何应用于以状态依赖的规则速率被软杀死的布朗扩散过程，并刻画相应的漂移变化。在扩散过程可逆的情形下，我们利用这一思想以及HJB半群弱对数凹性传播的最新结果，证明在势函数严格渐近凸的条件下，条件动力学在1-Wasserstein距离下满足一致于T>0的收缩性质。在相关的Feynman-Kac半群的一般遍历性条件下，我们随后建立具有大吸引域的QSD的存在性，以及条件半群在T趋于无穷时在1-Wasserstein距离下指数快速收敛到该QSD。最后，我们推导出相应的Q-过程的分布向其平衡态的指数快速收敛，同样是在1-Wasserstein距离下。

英文摘要

We develop a novel pathwise approach to study the convergence of the law of killed diffusion processes conditioned on non-absorption, towards a quasi-stationary distribution (QSD) as time goes to infinity. We start from the general observation that the dynamics of an absorbed Markov process conditioned upon survival up to time $T>0$ is the minimizer of the pathwise relative entropy with respect to its unconditioned dynamics, under a simple distributional constraint at that time; in other words, a Föllmer process. We then show how this result applies to a Brownian diffusion process softly-killed at a state-dependent regular rate, and characterize the associated drift change. In the case when the diffusion process is moreover reversible, we leverage this idea and recent results on the propagation of weak log-concavity of HJB semigroups to prove that, under strict asymptotic convexity of the potential, the conditioned dynamics satisfy a contractivity property in $1$-Wasserstein distance, uniformly in $T>0$. Under a general ergodicity condition on the associated Feynman-Kac semigroup, we then establish the existence of a QSD with a large domain of attraction, and the exponentially fast convergence to it of the conditioned semigroup in the $1$-Wasserstein distance as $T$ goes to infinity. Finally, we deduce the exponentially fast convergence, also in $1$-Wasserstein distance, of the law of the corresponding Q-process towards its equilibrium.

发表机构

• Centro de Modelamiento Matemático, Universidad de Chile（智利大学数学建模中心）
• IRL 2807 - CNRS（法国国家科学研究中心第2807国际研究实验室）
• Department of Mathematics, UCLA（加州大学洛杉矶分校数学系）

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