Superiority Of Symplectic Methods For Stochastic Hamiltonian System Via Asymptotic Error Distribution

The superiority of symplectic methods for stochastic Hamiltonian systems has been widely recognized, yet the probabilistic mechanism behind this superiority remains incompletely understood. This paper studies the superiority of symplectic methods from the perspective of the asymptotic error distribution, i.e., the limit distribution of normalized error. Focusing on stochastic Hamiltonian systems driven by additive noise, we obtain the asymptotic limit of the normalized error distribution of the $\theta$ method $(\theta\in[0,1])$ that is symplectic if and only if $\theta=\frac12$. By establishing upper bounds for the second-order moment of the asymptotic error distribution, we show that the midpoint method minimizes the error constant of the $\theta$ method for a large time horizon $T$. Furthermore, we take the linear stochastic oscillator as a test equation and investigate exact asymptotic error constants of several symplectic and non-symplectic methods. Our result suggests that in the long-time computation, the probability that the error deviates from zero decays exponentially faster for the symplectic methods than that for the non-symplectic ones.