Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp spectral gap of adaptive Langevin dynamics

Loïs Delande

TL;DR

The paper analyzes the adaptive Langevin dynamics through its degenerate Fokker-Planck generator in the semiclassical limit $h\to0$, establishing sharp spectral information linked to metastability. By relating the non-self-adjoint operator to semiclassical Witten Laplacians and introducing a tailored hypocoercivity framework with an auxiliary operator, it derives resolvent bounds and precise localization of the bottom of the spectrum. Central to the results is the construction of sharp Gaussian quasimodes via a WKB-style solution of the eikonal and transport equations, enabling explicit Eyring–Kramers type formulas in a two-well setting: the first nonzero eigenvalue scales as $\lambda= h e^{-S/h}$ with a computable prefactor depending on Hessians of $V$ at the wells and saddle. The methodology extends sharp semiclassical eigenvalue analysis to a degenerate, non-self-adjoint context and provides a clear mechanism for metastability and exit-time estimates for adaptive Langevin dynamics.

Abstract

We consider a degenerated Fokker-Planck type differential operator associated to an adaptive Langevin dynamic. We prove Eyring-Kramers formulas for the bottom of the spectrum of this operator in the low temperature regime. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.

Sharp spectral gap of adaptive Langevin dynamics

TL;DR

The paper analyzes the adaptive Langevin dynamics through its degenerate Fokker-Planck generator in the semiclassical limit , establishing sharp spectral information linked to metastability. By relating the non-self-adjoint operator to semiclassical Witten Laplacians and introducing a tailored hypocoercivity framework with an auxiliary operator, it derives resolvent bounds and precise localization of the bottom of the spectrum. Central to the results is the construction of sharp Gaussian quasimodes via a WKB-style solution of the eikonal and transport equations, enabling explicit Eyring–Kramers type formulas in a two-well setting: the first nonzero eigenvalue scales as with a computable prefactor depending on Hessians of at the wells and saddle. The methodology extends sharp semiclassical eigenvalue analysis to a degenerate, non-self-adjoint context and provides a clear mechanism for metastability and exit-time estimates for adaptive Langevin dynamics.

Abstract

We consider a degenerated Fokker-Planck type differential operator associated to an adaptive Langevin dynamic. We prove Eyring-Kramers formulas for the bottom of the spectrum of this operator in the low temperature regime. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.
Paper Structure (17 sections, 18 theorems, 228 equations, 1 figure)

This paper contains 17 sections, 18 theorems, 228 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivations.
  3. Statements.
  4. Hypocoercive estimates
  5. Witten Laplacians and an auxiliary operator.
  6. Boundedness of remaining terms.
  7. Hypocercivity result.
  8. Proof of Theorem \ref{['thm']}.
  9. Sharp quasimodes
  10. Solving the eikonal equation.
  11. Solving the transport equation.
  12. Simpler case of a two wells function
  13. Proof of Theorem \ref{['thm3']}.
  14. Some technical results
  15. Proof of Proposition \ref{['prop:accretive']}.
  16. ...and 2 more sections

Key Result

Proposition 1.1

The operator $P$ initially defined on $C^\infty_c(\mathbb R^{2d+1})$ admits a unique maximally accretive extension that we still denote by $(P,D(P))$, and we have $D(P) = \{u\in L^2(\mathbb R^{2d+1})\ |\ Pu\in L^2(\mathbb R^{2d+1})\}$.

Figures (1)

  • Figure 4.1: Representation of a typical two-wells Morse function

Theorems & Definitions (24)

  • Proposition 1.1
  • Lemma 1.2
  • Theorem 1
  • Remark 1.3
  • Theorem 2
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 14 more