Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation

TL;DR

This paper investigates the density convergence of a splitting AVF numerical scheme for stochastic Langevin equations, establishing smoothness, non-degeneracy, and optimal convergence rates using Malliavin calculus.

Contribution

It provides the first analysis of density function convergence for the splitting AVF scheme applied to stochastic Langevin equations, including smoothness and optimal convergence results.

Findings

Proves exponential integrability of solutions.

Establishes smoothness and non-degeneracy of the density.

Shows the density convergence rate matches the strong convergence rate.

Abstract

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. To deal with the non-globally monotone coefficient in the considered equation, we first present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donsker's delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme.