Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincar'e--Lions inequalities

2024-07-22

AI Summary

研究弱约束条件下的欠阻尼Langevin动力学长期行为，解决收敛速度量化问题。采用加权Poincaré-Lions不等式方法，针对空间均衡分布不满足标准Poincaré不等式的情况，提供L²范数下的显式估计。

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems

Abstract

We study the long-time convergence behavior of underdamped Langevin dynamics, when the spatial equilibrium satisfies a weighted Poincar'e inequality, with a general velocity distribution, which allows for fat-tail or subexponential potential energies, and provide constructive and fully explicit estimates in $mathrm{L}^2$-norm with $mathrm{L}^infty$ initial conditions. A key ingredient is a space-time weighted Poincar'e--Lions inequality, which in turn implies a weak Poincar'e--Lions inequality.

Problem

• Quantify convergence rate of underdamped Langevin dynamics
• Study weak confinement with fat-tail potential energies
• Provide explicit L²-norm estimates for L∞ initial data

Innovation

• Underdamped Langevin dynamics for weak confinement
• Weighted Poincaré inequality for fat-tail potentials
• Space-time weighted Poincaré-Lions inequality