Abstract

We first review some basic results on stochastic symplectic methods of stochastic Hamiltonian systems, including the  construction and analysis of stochastic symplectic integrators inclusive of stochastic generating functions, variational integrators, pseudo-symplectic methods, numerical ergodicity, and invariant measures. Then we characterize the probabilistic superiority of stochastic symplectic methods of stochastic Hamiltonian systems via large deviations principle of numerical observables and errors.