Stochastic differential equations (SDEs) are used to describe several real-life phenomena whose underlying dynamics depends on random fluctuations. This is the case, for example, of weather forecasts, turbulent diffusion or investment finance. In fact, SDEs provide a key tool for a mesoscopic approach to describe the effects of external environments to a physical model. In this talk we specically focus on the numerical discretization of stochastic Hamiltonian problem with additive noise, that are the most suitable candidate to conciliate the classic Hamiltonian mechanics with the non-differentiable Wiener process which describes the continuous innovative character of stochastic diffusion. Specically, our analysis involves stochastic Runge-Kutta methods obtained as a stochastic perturbation of symplectic Runge-Kutta methods, in order to understand their role in retaining invariance laws of the underlying dynamical system. In particular, we are interested in maintaining the linear drift visible in the expected Hamiltonian of the system. We give explanation to the preservation of this linear drift by means of a perturbative analysis, in terms of epsilon-expansions, being epsilon the amplitude of the stochastic part of the right-hand side. The presence of spurious terms growing in time and with epsilon is also visible and explained. Numerical tests confirm the theoretical analysis. References  A. Bazzani, Hamiltonian systems and Stochastic processes, Lecture Notes, University of Bologna (2018).  P.M. Burrage, K. Burrage, Low rankRunge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, Journal of Computational and Applied Mathematics, 236, 3920-3930 (2012).  P.M. Burrage, K. Burrage, Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algorithms 65, 519-532 (2014).  C. Chen, D. Cohen, R. D'Ambrosio, A. Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math 46, 27 (2020).  R. D'Ambrosio, G. Giordano, B. Paternoster, Numerical conservation issues for stochastic Hamiltonian problems, submitted.
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