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The Smoluchowski-Kramers approximation for a system with arbitrary friction depending on both state and distribution

Xueru Liu, Qianqian Jiang, Wei Wang

Abstract

A system of stochastic differential equations describing diffusive phenomena, which has arbitrary friction depending on both state and distribution is investigated. The Smoluchowski-Kramers approximation is seen to describe dynamics in the small mass limit. We obtain the limiting equation and, in particular, the addition drift terms that appear in the limiting equation are expressed in terms of the solutions to the Lyapunov matrix equation and Sylvester matrix equation. Furthermore, we provide the rate of convergence and extend the system to encompass more general interactions and noise.

The Smoluchowski-Kramers approximation for a system with arbitrary friction depending on both state and distribution

Abstract

A system of stochastic differential equations describing diffusive phenomena, which has arbitrary friction depending on both state and distribution is investigated. The Smoluchowski-Kramers approximation is seen to describe dynamics in the small mass limit. We obtain the limiting equation and, in particular, the addition drift terms that appear in the limiting equation are expressed in terms of the solutions to the Lyapunov matrix equation and Sylvester matrix equation. Furthermore, we provide the rate of convergence and extend the system to encompass more general interactions and noise.

Paper Structure

This paper contains 7 sections, 7 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Tightness and Well-posedness
  4. Tightness
  5. Well-posedness of equation (\ref{['maineq']})
  6. The rate of convergence
  7. Extension

Key Result

Lemma \oldthetheorem

Assume that assumptions a1-a4 hold. For every $T>0$, there exists a constant $C_{T}>0$ such that and for every $0\leq s\leq t\leq T$ and $\epsilon>0$, there exist a constant $C_{T}>0$ such that

Theorems & Definitions (20)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Remark \oldthetheorem
  • ...and 10 more