This is a preleminary work. Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many reversible overdamped Langevin dynamics which preserve the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.

翻译：这是一项初步工作。过阻尼朗之万动力学是可逆随机微分方程，常用于对高维空间中的概率测度进行采样，例如计算统计物理和贝叶斯推断中出现的测度。通过改变扩散系数，实际上存在无穷多个可逆过阻尼朗之万动力学，它们都能保持目标概率测度不变。这表明可以优化扩散系数以提高动力学的收敛速率，该速率由随机微分方程生成元的谱间隙来衡量。本文对该问题进行了分析研究，尤其得到了最优扩散系数的必要条件。我们还在某种均匀化极限下推导出了最优扩散的显式表达式。数值结果——既依赖于谱间隙问题的离散化，也依赖于随机动力学的蒙特卡洛模拟——展示了通过适当选择扩散系数所获得的采样质量的提升。