Before revision: We perform a numerical analysis of randomly perturbed Poisson systems } } . For the considered It\^o perturbation of Poisson differential equations , we show the longtime behavior of the energy and quadratic Casimirs for the exact solution.
After revision: We perform a numerical analysis of randomly perturbed Poisson systems } } . For the considered additive noise perturbation of such systems , we show the longtime behavior of the energy and quadratic Casimirs for the exact solution.