arXiv 2501.136552025-01math.PR

遍历McKean随机微分方程的线性化及其应用

Linearization of ergodic McKean SDEs and applications

Grigorios A. Pavliotis, Andrea Zanoni

AI总结：

针对具有唯一平稳态的McKean随机微分方程，提出将卷积项分布替换为不变测度的线性化方法，证明其分布指数收敛，还提出渐近无偏的线性化最大似然估计并研究粒子系统的联合扩散-平均场极限。

摘要：

本文研究具有唯一平稳态的McKean随机微分方程及其对应的McKean-Vlasov偏微分方程，考察将卷积项中的过程分布替换为唯一不变测度后得到的线性化Itô扩散过程。我们证明，非线性McKean过程的分布在相对熵和Wasserstein距离下，均随时间指数快速收敛到该线性化过程的分布。我们在全空间和环面两种情形下研究了该问题。随后，我们展示了如何利用所得的线性（McKean意义下）马尔可夫过程，分析原始非线性非局部动力学中依赖于其长时间行为的性质。特别地，我们针对非线性过程提出了一种渐近无偏的线性化最大似然估计量，并研究了底层相互作用粒子系统的联合扩散-平均场极限。

英文摘要：

In this article, we consider McKean stochastic differential equations, as well as their corresponding McKean-Vlasov partial differential equations, which admit a unique stationary state, and we study the linearized Itô diffusion process that is obtained by replacing the law of the process in the convolution term with the unique invariant measure. We show that the law of the nonlinear McKean process converges to the law of this linearized process exponentially fast in time, both in relative entropy and in Wasserstein distance. We study the problem in both the whole space and the torus. We then show how we can employ the resulting linear (in the sense of McKean) Markov process to analyze properties of the original nonlinear and nonlocal dynamics that depend on their long-time behavior. In particular, we propose a linearized maximum likelihood estimator for the nonlinear process which is asymptotically unbiased, and we study the joint diffusive-mean field limit of the underlying interacting particle system.

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