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The small mass limit for a McKean-Vlasov equation with state-dependent friction

Chungang Shi, Mengmeng Wang, Yan Lv, Wei Wang

Abstract

The small mass limit is derived for a McKean-Vlasov equation with state-dependent friction in $d$-dimensional space. By applying the averaging approach to a non-autonomous slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained.

The small mass limit for a McKean-Vlasov equation with state-dependent friction

Abstract

The small mass limit is derived for a McKean-Vlasov equation with state-dependent friction in -dimensional space. By applying the averaging approach to a non-autonomous slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained.
Paper Structure (3 sections, 10 theorems, 152 equations)

This paper contains 3 sections, 10 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminary and main result
  3. An averaging result: $\epsilon\to0$

Key Result

Lemma 2.1

BBH Let $I\subset\mathbb{R}$ an open interval with $t_{0}\in I$, $A\in \mathbb{C}^{n\times n}, B\in \mathbb{C}^{m\times m}, C\in \mathcal{C}(I,\mathbb{C}^{n\times n})$ and $D\in \mathbb{C}^{m\times n}$. The differential Sylvester equation has the unique solution

Theorems & Definitions (19)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 9 more