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

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Abstract

We provide a new explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance. To achieve this, we first prove a Poincar'{e}-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our new estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincar'{e} constant $m \ll 1$, our new estimate offers the decay rate of $\mathcal{O}(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster compared to $\mathcal{O}(m)$ for the overdamped Langevin dynamics.

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Lu

Jianfeng Lu

James B. Duke Distinguished Professor

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science, machine learning, and other related fields.

More specifically, his current research focuses include:
High dimensional PDEs; generative models and sampling methods; control and reinforcement learning; electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis.


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