In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. We first show the existence of the density function of the numerical solution by proving its exponential integrability property, Malliavin differentiability and the almost surely non-degeneracy of the associated Malliavin covariance matrix. Then the smoothness of the density function is obtained through a lower bound estimate of the smallest eigenvalue of the corresponding Malliavin covariance matrix. Meanwhile, we derive the optimal strong convergence rate in every Malliavin–Sobolev norm of the numerical solution via Malliavin calculus. Combining the strong convergence result and the smoothness of the density functions, we prove that the convergence order of the density function of the numerical scheme coincides with its strong convergence order. 

　　Publication: 

　　Mathematics of Computation, Volume 91 (2022), pp: 2283-2333. 

　　Author: 

　　Jianbo Cui 

　　Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Republic of China 

　　Email address: jianbo.cui@polyu.edu.hk 

　　Jialin Hong 

　　LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China 

　　Email address: hjl@lsec.cc.ac.cn 

　　Derui Sheng  

　　LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China; and School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China 

　　Email address: sdr@lsec.cc.ac.cn