On the perturbative analysis of the time-discretization for stochastic Hamiltonian problems

Stochastic Differential Equations (SDEs) are excellent models used to describe several natu-ral and real-life phenomena, when they are subject to random perturbations. This is the case, for example, of weather forecasts, turbulent diffusion or investment finance. Indeed, SDEs provide a key tool for a "mesoscopic" approach to describe the effects of external environments to a physical model. The irreversible character of a stochastic dynamics destroys the idea of isolate systems, since the particles are repeatedly influenced by small unpredictable perturbations of the external environment. In this talk, specifically, we focus on the study on the dynamics of stochastic Hamiltonian problems because they represent a suitable candidate to conciliate the canonical character of the evolution equations, with the non-differentiability of the Wiener process, that describes the continuous innovative character of stochastic diffusion. Our analysis focuses on the study of stochastic Runge-Kutta methods developed by Burrage and Burrage, obtained through a stochastic perturbation of symplectic Runge-Kutta methods, in order to understand if they maintain the linear drift visible in the expected Hamiltonian of the system. In particular we observe that stochastic Runge-Kutta methods exhibit a remarkable error that increases with the parameter ε, describing the amplitude of the diffusive part of the problem. Through a perturbative analysis, in terms of ε expansions, we investigate the reason of this behaviour and exhibit the presence of a secular term ε*sqrt(t) that destroys the overall conservation accuracy. The theoretical results are also confirmed by selected numerical experiments.