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Exponential contractivity and propagation of chaos for Langevin dynamics of McKean-Vlasov type with Lévy noises

Yao Liu, Jian Wang, Meng-ge Zhang

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 $(X_t,Y_t)_{t\ge0}$ 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)μ^X_t(dz)-γY_t\right)dt+dL_t,\quad μ^X_t={\rm Law}(X_t),\end{array}\right. \end{equation*} where $γ>0$, $b:\mathbb{R}^d\rightarrow\mathbb{R}^d$ 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 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.

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 -Wasserstein distance for the following Langevin dynamic of McKean-Vlasov type on : \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)μ^X_t(dz)-γY_t\right)dt+dL_t,\quad μ^X_t={\rm Law}(X_t),\end{array}\right. \end{equation*} where , and are two globally Lipschitz continuous functions, and is an -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 -Wasserstein distance as well as with explicit bounds.
Paper Structure (15 sections, 15 theorems, 175 equations)

This paper contains 15 sections, 15 theorems, 175 equations.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Overview on the existing results and approaches
  4. Strong solution of Langevin dynamics of McKean-Vlasov type with jumps and coupling process of their decoupled version
  5. Uniqueness of strong solution to the SDE \ref{['Equation']}
  6. Coupling operator for the decoupled SDE \ref{['decoupled1']}
  7. Coupling process for the decoupled SDE \ref{['decoupled1']}
  8. Proof of Theorem \ref{['Main theorem']}
  9. Proof of Theorem \ref{['chaos']}
  10. Coupling process for the particle system \ref{['meanfield']} and the decoupled SDE \ref{['decoupledN']}
  11. Proof of Theorem \ref{['chaos']}
  12. Appendix
  13. Strong solution to McKean-Vlasov SDEs with Lévy noises
  14. Comparison of two metrics $r_s((x,y),(x',y'))$ and $r_l((x,y),(x',y'))$
  15. Declarations

Key Result

Theorem 1.1

Suppose that Assumptions (A0)--(A2) hold, and the constants involved in Assumption (A1) satisfy Then there exists a constant $C_{\tilde{b}}>0$ such that, when the constant in Assumption (A2) fulfills $L_{\tilde{b}}\le C_{\tilde{b}}$, there are constants $\lambda,\,C_1>0$ so that for any $t>0$ and $\mu,\,\bar{\mu}\in\mathscr{P}_1(\mathbb R^{2d})$ where $\mu_t$ and $\bar{\mu}_t$ are the laws of the

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 22 more