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Drift-preserving numerical integrators for stochastic Poisson systems

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D. Cohen and G. Vilmart

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Abstract. We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long time behavior of the energy and quadratic Casimirs for the exact solution. We then propose and analyze a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence one, weak order of convergence two. These properties are illustrated with numerical experiments.

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Key Words. Stochastic differential equations, Stochastic Hamiltonian systems, Stochastic Poisson systems, Energy, Casimir, Trace formula, Numerical schemes, Strong convergence, Weak convergence.

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