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Uniform-in-time diffusion approximations for multiscale stochastic systems

Longjie Xie, Xicheng Zhang

Abstract

This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits $\eps\to0$ and $t\to\infty$ for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: {\it (i)} a uniform-in-time averaging principle for fast-slow systems; {\it (ii)} a uniform Smoluchowski--Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and {\it (iii)} the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.

Uniform-in-time diffusion approximations for multiscale stochastic systems

Abstract

This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits and for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: {\it (i)} a uniform-in-time averaging principle for fast-slow systems; {\it (ii)} a uniform Smoluchowski--Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and {\it (iii)} the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.

Paper Structure

This paper contains 44 sections, 25 theorems, 393 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results and contributions
  4. Notation: Polynomial weights and weighted Hölder spaces
  5. Hypotheses.
  6. Applications
  7. Uniform-in-time averaging principle (Section 5)
  8. Uniform Smoluchowski--Kramers approximation with state-dependent friction (Section 6)
  9. Periodic homogenization with distributional drifts (Section 7)
  10. Overview of the proof and organization of the paper
  11. Preliminaries: Banach-valued weighted Hölder spaces
  12. Parameter-dependent Poisson and Kolmogorov equations
  13. Parameterized SDE and standing assumptions
  14. Poisson equation on the whole space with a parameter
  15. Infinite-horizon Kolmogorov equation with a parameter
  16. ...and 29 more sections

Key Result

Theorem 1.1

Assume that ${\bf (H)}$ hold. In addition, suppose that for some $\alpha,\beta\in(0,1]$, Then for any $\gamma\in(\beta,2)$ and $(\rho_0,\omega_0)\in{\mathscr W}_d\times{\mathscr W}_\vartheta$, there exist a weight $\varrho\in{\mathscr W}_{d+\vartheta}$ such that for all $\varphi\in{\mathbf C}^\alpha_{\rho_0}{\mathbf C}^\gamma_{\omega_0}$, $\varepsilon\in(0,1)$, $z=(x,y)\in{\mathbb R}^d\ where $\b

Theorems & Definitions (60)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 50 more