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On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics

Yu Cao, Jianfeng Lu, Lihan Wang

Abstract

We provide a refined explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance, based on a framework developed in [1]. To achieve this, we first prove a Poincaré-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincaré constant $m \ll 1$, our estimate shows the decay rate of $O(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster than $m$ for the overdamped Langevi dynamics.

On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics

Abstract

We provide a refined explicit estimate of exponential decay rate of underdamped Langevin dynamics in distance, based on a framework developed in [1]. To achieve this, we first prove a Poincaré-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincaré constant , our estimate shows the decay rate of after optimizing the choice of friction coefficient, which is much faster than for the overdamped Langevi dynamics.

Paper Structure

This paper contains 9 sections, 12 theorems, 115 equations, 1 table.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. A literature review and comparison
  4. Proofs
  5. The decay rate for isotropic quadratic potential
  6. The DMS hypocoercive estimation
  7. Revisiting the DMS hypocoercive estimation in $L^2(\rho_{\infty})$
  8. Asymptotic analysis of the decay rate
  9. Proofs of the Propositions in Appendix

Key Result

Theorem 1

Under Assumptions assump:poincare, assump:hessian, and assump:spectral, there exist a constant $\nu > 0$ and universal constants $C,c$ independent of all parameters such that, for every $f(t,x,v)$ satisfying the backward Kolmogorov equation eqn::opL with initial condition $f_0 \in L^2(\mu;H^1_\kapp we have, for every $t\in (0,\infty)$, Moreover, $\nu$ can be made explicit as with some constant

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 14 more