On the long-time integration of stochastic gradient systems

This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize h→0) convergence behaviour of the error of finite-time averages. Recently, it has been demonstrated, by study of Fokker.Planck operators, that a non-Markovian numerical method generates approximations in the long-time limit with higher accuracy order (second order) than would be expected from its weak convergence analysis (finite-time averages are first-order accurate). In this article, we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to second order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler.Maruyama method, the popular second-order Heun method, and the non-Markovian method.