How do Monte Carlo estimates affect stochastic geometric numerical integration?

In this work, we investigate the numerical conservation of characteristic properties of stochastic Hamiltonian problems driven by additive noise under time discretizations, in the spirit of stochastic geometric numerical integration. In particular, we aim to understand how Monte Carlo and multilevel Monte Carlo estimators for the expected values eventually affect the conservation properties of the numerical scheme. Specifically, our analysis is focused on the role of random number generation in the conservation of invariant laws for stochastic Hamiltonian problems under time discretization by the drift-preserving numerical scheme introduced in C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, [Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46(2) (2020), pp. 27.] and the stochastic perturbation of a symplectic Runge–Kutta method, introduced in K. Burrage and P.M. Burrage, [Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236(16) (2012), pp. 3920–3930.]. The numerical evidence confirms the aforementioned theoretical analysis.