Asymptotically-Preserving Large Deviations Principles by Stochastic Symplectic Methods for a Linear Stochastic Oscillator

It is well known that symplectic methods have been rigorously shown to be superior to nonsymplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the superiority of stochastic symplectic methods by means of the large deviations principle. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Considering that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the Gärtner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but nonsymplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the “hitting probability" of the mean position and mean velocity of the stochastic oscillator. Finally, numerical experiments are performed to show the superiority of stochastic symplectic methods in computing the large deviations rate functions. To the best of our knowledge, this is the first result about applying the large deviations principle to reveal the superiority of stochastic symplectic methods compared with nonsymplectic ones in the existing literature.