Density  of Langevin Equation and Its Approximation

This chapter focuses on the existence, smoothness, and approximation of densities of solutions to stochastic ordinary differential equations under H \({\ddot{\text {o}}}\) rmander’s condition. Special attention is paid to the Langevin equation with polynomial growth coefficients, which is highly relevant in multilevel Monte Carlo simulation, Langevin Monte Carlo algorithms, and machine learning. We introduce a splitting averaged vector field method for the Langevin equation and investigate the existence, smoothness, and convergence of the density of its numerical solutions.