Computing ergodic limits for Langevin equations

Abstract

To evaluate averages with respect to the invariant law for Langevin-type equations, one has to integrate a system over long time intervals, especially when dissipation is small. This is a challenging problem from the computational point of view. Since coefficients of the Langevin equations are typically not globally Lipschitz, the difficulties are redoubled. In the nonglobal Lipschitz case we observe an exploding behavior of some approximate trajectories due to many usual numerical methods. In [G.N. Milstein, M.V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal. 43 (2005) 1139–1154] we propose and justify a concept which, in principle, allows us to apply any numerical method to stochastic differential equations with nonglobally Lipschitz coefficients. Here quasi-symplectic integrators from [G.N. Milstein, M.V. Tretyakov, Quasi-symplectic methods for Langevin-type equations, IMA J. Numer. Anal. 23 (2003) 593–626] (they are the most appropriate methods for solving Langevin-type equations, especially when damping is small) and the concept of [G.N. Milstein, M.V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal. 43 (2005) 1139–1154] are applied for calculation of ergodic limits. Special attention is paid to the case when the invariant measure is known (e.g., when this measure is Gibbsian). Some computer experiments with three model problems (Van der Pol’s equation with additive noise, one-dimensional arrays of oscillators in thermal equilibrium, and a physical pendulum with linear friction and additive noise) are presented.