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The Smoluchowski-Kramers approximation for a McKean-Vlasov equation subject to environmental noise with state-dependent friction

Chungang Shi, Yan Lv, Wei Wang

Abstract

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

The Smoluchowski-Kramers approximation for a McKean-Vlasov equation subject to environmental noise with state-dependent friction

Abstract

The small mass limit is derived for a McKean-Vlasov equation subject to environmental noise with state-dependent friction. By applying the averaging approach to a non-autonomous stochastic slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained.
Paper Structure (3 sections, 7 theorems, 144 equations)

This paper contains 3 sections, 7 theorems, 144 equations.

Table of Contents

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

Key Result

Theorem 2.2

Under the assumptions $(\mathbf{H_{1}})$-$(\mathbf{H_{4}})$, for any $T>s_{0}>0$, $\rho_{t}^{\epsilon}(x)\triangleq \int_{\mathbb{R}^{d}}\mu_{t}^{\epsilon}(x,v)dv$ converges weakly to $\rho_{t}$ for $s_{0}\leq t\leq T$ as $\epsilon\to0$ with which corresponds to the following SDE where $\tilde{X}_{t}$ is a version of $X_{t}$, $\tilde{E}$ is the expectation with respect to the distribution of $\t

Theorems & Definitions (15)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 5 more