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

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.