Preserving structures of stochastic differential equations along numerical solutions

The aim of this talk is the analysis of the long-term behavior of stochastic linear multistep methods applied to a family of second order stochastic differential equations, modeling a stochastic damped oscillator, i.e. describing the position of a particle subject to the deterministic forcing and a random forcing whose global dynamics exhibits damped oscil- lations. In particular, the talk focuses on preserving long-term statistics related to such a dynamics; the velocity, in the stationary regime, is distributed as a Gaussian variable and uncorrelated with the position [1]. By computing the solution of a very simple ma- trix equality, we a-priori compute the long-term statistics characterizing the numerical dynamics and analyze the behaviour of a selection of methods [2]. References [1] K. Burrage and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput. 29 (1) (2007), 245–264. [2] R. D’Ambrosio, M. Moccaldi and B. Paternoster, Long-term preservation of invariance laws by stochastic multistep methods, submitted.