Exponential contractivity and propagation of chaos for Langevin dynamics of McKean-Vlasov type with L\'evy noises

TL;DR

This paper establishes explicit exponential contraction rates and propagation of chaos for a class of Langevin McKean-Vlasov dynamics driven by Lévy noises, using a novel coupling approach and distance function.

Contribution

It introduces a refined coupling method and a new distance function to analyze exponential convergence and chaos propagation in Lévy-driven Langevin dynamics.

Findings

Explicit exponential contraction rates in Wasserstein distance.

Uniform propagation of chaos with explicit bounds.

Application to mean-field particle systems with Lévy noise.

Abstract

By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for L\'evy processes, we obtain explicit exponential contraction rates in terms of the standard $L^1$-Wasserstein distance for the following Langevin dynamic $(X_t,Y_t)_{t\ge0}$ of McKean-Vlasov type on $\mathbb{R}^{2d}$: \begin{equation*}\left\{\begin{array}{l} dX_t=Y_tdt,\\ 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={\rm Law}(X_t),\end{array}\right. \end{equation*} 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 $(L_t)_{t\ge0}$ is an $\mathbb{R}^d$-valued pure jump L\'evy 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 L\'evy noises in the standard $L^1$-Wasserstein distance as well as with explicit bounds.