Ergodicity Of Langevin Diffusion Process

Ergodicity of underdamped Langevin diffusion process has important applications in numerically solving non-convex optimization problems of machine learning.For the ergodicity of Langevin diffusion process(sometimes called Langevin dynamics),people mainly concern(Problem I)Under the Wasserstein metric,what is the rate at which the probabilitydistribution of the underdamped Langevin diffusion process converges to its sta-tionary distribution?(Problem II)In what way does the above convergence rate depend on the parameters of the non-convex optimization problem?This paper mainly summarizes and supplements some results of the recent aca-demic paper " Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization"[1]on the application of the ergodicity of the Langevin diffusion process to solve a class of optimization problems.The main concern here is the ergod-icity of the underdamped Langevin diffusion process(Langevin dynamics)when the objective function in the optimization problem is a quadratic function.The main idea of studying the problems(1)and(2)is to transform the problem into a problem of es-timating the spectral norm of a correlation matrix.This paper further supplements the proof of some related conclusions in the paper mentioned above.