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Large Deviations for Slow-Fast Mean-Field Diffusions

Wei Hong, Wei Liu, Shiyuan Yang

TL;DR

The paper develops a rigorous large deviation principle for slow-fast mean-field diffusions where both slow and fast components influence the law of the system. By introducing functional occupation measures and a viable-pair framework with feedback controls, the authors overcome obstacles posed by the time-marginal law of the fast process and derive the Laplace-principle bounds. They present an explicit rate function $I(\varphi)$ in terms of a positive definite operator $Q$ and the averaged drift $\bar{b}$, highlighting the role of the invariant measure $\nu$ of the fast dynamics. This work provides a comprehensive multiscale large deviations framework with potential applications to efficient importance sampling and rare-event analysis in complex mean-field systems.

Abstract

The aim of this paper is to investigate the large deviations for a class of slow-fast mean-field diffusions, which extends some existing results to the case where the laws of fast process are also involved in the slow component. Due to the perturbations of fast process and its time marginal law, one cannot prove the large deviations based on verifying the powerful weak convergence criterion directly. To overcome this problem, we employ the functional occupation measure, which combined with the notion of the viable pair and the controls of feedback form to characterize the limits of controlled sequences and justify the upper and lower bounds of Laplace principle. As a consequence, the explicit representation formula of the rate function for large deviations is also presented.

Large Deviations for Slow-Fast Mean-Field Diffusions

TL;DR

The paper develops a rigorous large deviation principle for slow-fast mean-field diffusions where both slow and fast components influence the law of the system. By introducing functional occupation measures and a viable-pair framework with feedback controls, the authors overcome obstacles posed by the time-marginal law of the fast process and derive the Laplace-principle bounds. They present an explicit rate function in terms of a positive definite operator and the averaged drift , highlighting the role of the invariant measure of the fast dynamics. This work provides a comprehensive multiscale large deviations framework with potential applications to efficient importance sampling and rare-event analysis in complex mean-field systems.

Abstract

The aim of this paper is to investigate the large deviations for a class of slow-fast mean-field diffusions, which extends some existing results to the case where the laws of fast process are also involved in the slow component. Due to the perturbations of fast process and its time marginal law, one cannot prove the large deviations based on verifying the powerful weak convergence criterion directly. To overcome this problem, we employ the functional occupation measure, which combined with the notion of the viable pair and the controls of feedback form to characterize the limits of controlled sequences and justify the upper and lower bounds of Laplace principle. As a consequence, the explicit representation formula of the rate function for large deviations is also presented.
Paper Structure (14 sections, 14 theorems, 186 equations)

This paper contains 14 sections, 14 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Main assumptions
  3. Main result on LDP
  4. Weak convergence approach
  5. Main result
  6. Preliminaries
  7. Some a priori estimates
  8. Tightness of controlled processes
  9. Proof of LDP
  10. Existence of viable pair
  11. Lower bound of Laplace principle
  12. Compactness of level sets of $I(\cdot)$
  13. Upper bound of Laplace principle
  14. Appendix

Key Result

Theorem 3.1

Suppose that the assumptions $(\mathbf{A_1})$-$(\mathbf{A_2})$ hold and the scale $\varepsilon=\varepsilon(\delta)$ satisfies $\lim_{\delta\to0} \varepsilon/\delta = 0$. Then $\{X^\delta\}_{\delta>0}$ satisfies the LDP with the good rate function $I$ given by with the convention that the infimum over the empty set is $\infty$, where $\Phi$ is defined by $(2.14)$ and $\bar{X}$ is the solution of $

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Definition 3.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • proof
  • ...and 24 more