Exponential Contractivity and Propagation of Chaos for Langevin Dynamics of McKean-Vlasov Type with Lévy Noises

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-03-15 DOI:10.1007/s11118-024-10130-y

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引用次数: 0

Abstract

By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for Lévy processes, we obtain explicit exponential contraction rates in terms of the standard $L^1$ -Wasserstein distance for the following Langevin dynamic $(X_t,Y_t)_{t\ge 0}$ of McKean-Vlasov type on $\mathbb R^{2d}$ : $$\begin{aligned} \left\{ \begin{array}{l} dX_t=Y_t\,dt,\\ dY_t=\left( b(X_t)+\displaystyle \int _{\mathbb R^d}\tilde{b}(X_t,z)\,\mu ^X_t(dz)-{\gamma }Y_t\right) \,dt+dL_t,\quad \mu ^X_t=\textrm{Law}(X_t), \end{array} \right. \end{aligned}$$ where ${\gamma }>0$ , $b:\mathbb R^d\rightarrow \mathbb R^d$ and $\tilde{b}:\mathbb R^{2d}\rightarrow \mathbb R^d$ are two globally Lipschitz continuous functions, and $(L_t)_{t\ge 0}$ is an $\mathbb R^d$ -valued pure jump Lévy process. The proof is also based on a novel distance function, which is designed according to the distance of the marginals associated with the constructed coupling process. Furthermore, by applying the coupling technique above with some modifications, we also provide the propagation of chaos uniformly in time for the corresponding mean-field interacting particle systems with Lévy noises in the standard $L^1$ -Wasserstein distance as well as with explicit bounds.

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带列维噪声的麦金-弗拉索夫型兰万动力学的指数收缩性和混沌传播

摘要通过概率耦合方法（该方法将新的精炼基本耦合与莱维过程的同步耦合结合在一起），我们为 $\mathbb R^{2d}$ 上 McKean-Vlasov 类型的以下朗格文动态 $(X_t,Y_t)_{t\ge 0}$ 得到了以标准 $L^1$ -Wasserstein 距离表示的明确指数收缩率： $$\begin{aligned}\dX_t=Y_t\,dt,dY_t=left( b(X_t)+\displaystyle int _{\mathbb R^d}\tilde{b}(X_t,z)\,\mu ^X_t(dz)-{\gamma }Y_t\right) \,dt+dL_t,\quad \mu ^X_t=\textrm{Law}(X_t), \end{array}.\right.\end{aligned}$$ where ${\gamma }>0$ , $b:\mathbb R^d\rightarrow \mathbb R^d$ and$\tilde{b}：\是两个全局李普齐兹连续函数，并且（(L_t)_{t\ge 0}）是一个（\mathbb R^d\）-值的纯跳跃李维过程。证明还基于一个新颖的距离函数，该函数是根据与所构建的耦合过程相关的边际的距离设计的。此外，通过应用上述耦合技术并进行一些修改，我们还提供了在标准 \(L^1$ -Wasserstein 距离下，具有莱维噪声的相应均场相互作用粒子系统在时间上均匀的混沌传播以及显式边界。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.