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The Smoluchowski-Kramers approximation with distribution-dependent potential and highly oscillating force

Chungang Shi, Wei Wang

Abstract

An approximation is derived for a Langevin equation with distribution-dependent potential and state-dependent, randomly fast oscillation. By some estimates and a diffusion approximation the limiting equation is shown to be distribution-dependent stochastic differential equation (SDEs) driven by white noise.

The Smoluchowski-Kramers approximation with distribution-dependent potential and highly oscillating force

Abstract

An approximation is derived for a Langevin equation with distribution-dependent potential and state-dependent, randomly fast oscillation. By some estimates and a diffusion approximation the limiting equation is shown to be distribution-dependent stochastic differential equation (SDEs) driven by white noise.
Paper Structure (4 sections, 6 theorems, 97 equations)

This paper contains 4 sections, 6 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminary and main result
  3. Tightness of solutions
  4. Diffusion approximation

Key Result

Theorem 2.1

Assume $(\mathbf{H_{1}})$-$(\mathbf{H_{3}})$ hold, for any $T>0$, the solution $x^{\epsilon}$ of equation (equ:main) converges in distribution as $\epsilon\to0$ to $x(t)$ in space $C(0,T;\mathbb{R}^{d})$ with $x$ satisfying where $\Sigma=\mathbb{E}(\tilde{\mathbb{E}}\eta^{\epsilon}(t,x^{\epsilon}(t))\otimes\tilde{\mathbb{E}}\eta^{\epsilon}(t,x^{\epsilon}(t)))$, $\beta=-m'(0)\geq0$ and $B(t)$ is a

Theorems & Definitions (8)

  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.1