Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Empirical approximation to invariant measures of non-degenerate McKean-Vlasov dynamics

Wenjing Cao, Kai Du

Abstract

This paper studies the approximation of invariant measures of McKean-Vlasov dynamics with non-degenerate additive noise. While prior findings necessitated a strong monotonicity condition on the McKean-Vlasov process, we expand these results to encompass dissipative and weak interaction scenarios. Utilizing a reflection coupling technique, we prove that the empirical measures of the McKean-Vlasov process and its path-dependent counterpart can converge to the invariant measure in the Wasserstein metric. The Curie-Weiss mean-field lattice model serves as a numerical example to illustrate empirical approximation.

Empirical approximation to invariant measures of non-degenerate McKean-Vlasov dynamics

Abstract

This paper studies the approximation of invariant measures of McKean-Vlasov dynamics with non-degenerate additive noise. While prior findings necessitated a strong monotonicity condition on the McKean-Vlasov process, we expand these results to encompass dissipative and weak interaction scenarios. Utilizing a reflection coupling technique, we prove that the empirical measures of the McKean-Vlasov process and its path-dependent counterpart can converge to the invariant measure in the Wasserstein metric. The Curie-Weiss mean-field lattice model serves as a numerical example to illustrate empirical approximation.
Paper Structure (10 sections, 10 theorems, 118 equations, 1 figure)

This paper contains 10 sections, 10 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Convergence of empirical measures
  4. Convergence of weighted empirical measures
  5. Proof of Theorems \ref{['DistributionDependent']} and \ref{['PathDependent']}
  6. Distribution-dependent equations
  7. Path-dependent equations
  8. Proof of Theorem \ref{['WeightedPathDependent']}
  9. Wasserstein convergence for empirical measures of Markov processes
  10. Numerical experiments

Key Result

Theorem 2.1

Under Assumptions dissi, weaki and Lq, for any solution $X$ of the distribution-dependent SDE sde1, we have where $0<\varepsilon<\min\{\frac{1}{d}(1-\frac{1}{q}),\frac{1}{2}(1-\frac{1}{q})\}$.

Figures (1)

  • Figure 1: Expectation of Wasserstein distances between simulated empirical measures $\mathcal{E}_t(\hat{Z})$ and invariant measure $\mu^*$ of Curie Weiss mean-field lattice model \ref{['Curie']}.

Theorems & Definitions (18)

  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Remark 3.1
  • Remark 5.1
  • Proposition 5.1
  • Remark 5.2
  • Remark 5.3
  • ...and 8 more