论文标题
具有乘法噪声的未调整的langevin算法：总变化和瓦斯汀边界
Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds
论文作者
论文摘要
在本文中，我们关注与具有乘法扩散项（非恒定扩散系数）的沿寿命扩散相关的非反应界限。更确切地说，本文的目的是控制标准Euler方案的距离，以减小步骤（通常称为Monte Carlo文献中未经调整的Langevin算法}）到这种厄基德扩散的不变分布。在适当的lyapunov设置以及在扩散系数上的{统一}椭圆度假设下，我们在乘法和加法和框架中建立（或改善）总变化的界限和$ l^1 $ -Wasserstein的距离。这些界限依赖于使用{随机分析}适应降低步骤设置的弱误差扩展。
In this paper, we focus on non-asymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (non-constant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step ({usually called Unadjusted Langevin Algorithm in the Monte Carlo literature}) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under {uniform} ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for Total Variation and $L^1$-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using {Stochastic Analysis} adapted to decreasing step setting.
