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Learning Equilibrium Fluctuation Expansions from Overdamped Langevin Dynamics

Lin Wang, Zhengyan Wu

Abstract

We study higher-order small-noise fluctuation expansions for the overdamped Langevin dynamics in a quartic double-well potential. Assuming that the initial data admits a suitable expansion structure, we obtain a strong dynamical expansion of the trajectories, as well as an expansion of the laws with respect to smooth observables. We then investigate the long-time behavior of the expansion coefficients. In the scalar case $d=1$, each coefficient converges exponentially fast to a finite limit as $t\to\infty$. In contrast, for $d\ge 2$, the fluctuation expansion coefficients reflect the degeneracy of the manifold of minima, which in general prevents the existence of a finite long-time limit. Furthermore, by combining a multi-level induction with combinatorial arguments, we derive a recursive formula for the fluctuation expansion coefficients. This recursion shows that the long-time limits of these dynamical expansion coefficients coincide with those arising from the corresponding equilibrium expansions.

Learning Equilibrium Fluctuation Expansions from Overdamped Langevin Dynamics

Abstract

We study higher-order small-noise fluctuation expansions for the overdamped Langevin dynamics in a quartic double-well potential. Assuming that the initial data admits a suitable expansion structure, we obtain a strong dynamical expansion of the trajectories, as well as an expansion of the laws with respect to smooth observables. We then investigate the long-time behavior of the expansion coefficients. In the scalar case , each coefficient converges exponentially fast to a finite limit as . In contrast, for , the fluctuation expansion coefficients reflect the degeneracy of the manifold of minima, which in general prevents the existence of a finite long-time limit. Furthermore, by combining a multi-level induction with combinatorial arguments, we derive a recursive formula for the fluctuation expansion coefficients. This recursion shows that the long-time limits of these dynamical expansion coefficients coincide with those arising from the corresponding equilibrium expansions.

Paper Structure

This paper contains 15 sections, 9 theorems, 275 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Overview of the approach
  4. Generalizations
  5. Background and applications
  6. Comments on the literature
  7. Structure of the paper
  8. Dynamical asymptotic expansion
  9. $L^{2p}$-Estimates for the expansion coefficients
  10. Convergence in $L^{2p}$
  11. Weak asymptotic expansion
  12. Long-time behavior of the fluctuation expansion coefficients
  13. Scalar-valued case
  14. Vector-valued case
  15. Connection with expansions in equilibrium

Key Result

Proposition 1.1

Let $\varepsilon\in (0,1)$ and assume that $X_\varepsilon$ is the solution of SDE-0 with initial data $\xi_{\varepsilon}$ satisfying Assumption A1. Let $(\bar{X}_m)_{0\leqslant m\leqslant n}$ be the solution of eq-barX0-intro, eq-barX1-intro, eq-barXm-intro, respectively. We define the remainder ter Then for each $p\geqslant1$ and $t>0$, there exists a constant $C(p,t,n)>0$, independent of $\varep

Figures (1)

  • Figure 1: Schematic picture of the nested induction on $S_{n,i}$ in the case $n=4$. An arrow from $A$ to $B$ indicates that $A$ appears as a forcing term in the evolution equation for $B$.

Theorems & Definitions (12)

  • Proposition 1.1: Dynamical fluctuation expansion
  • Proposition 1.2: Weak expansion
  • Theorem 1.3: Long-time behavior in the scalar case
  • Theorem 1.4: Convergence rates of the long-time limits
  • Theorem 1.5: Identification of coefficients, Proposition \ref{['prop-invariant-exp']}, Theorem \ref{['thm-consistency']}
  • Theorem 1.6: Non-convergence in the vector case
  • Proposition 2.1
  • proof
  • Proposition 5.1
  • proof
  • ...and 2 more