The superiority of stochastic symplectic methods over non-symplectic counterparts has been verified by plenty of numerical experiments, especially in capturing the asymptotic behaviour of the underlying solution process. How can one theoretically explain this superiority? This paper gives an answer to this problem from the perspective of the law of iterated logarithm, taking the linear stochastic Hamiltonian system in Hilbert space as a test model. The main contribution is twofold. First, by fully utilizing the time-change theorem for martingales and the Borell--TIS inequality, we prove that the upper limit of the exact solution with a specific scaling function almost surely equals some non-zero constant, thus confirming the validity of the law of iterated logarithm. Second, we prove that stochastic symplectic fully discrete methods asymptotically preserve the law of iterated logarithm, but non-symplectic ones do not. This reveals the good ability of stochastic symplectic methods in characterizing the almost sure asymptotic growth of the utmost fluctuation of the underlying solution process. Applications of our results to the linear stochastic oscillator and the linear stochastic Schrodinger equation are also presented.

翻译：随机辛方法相对于非辛方法在捕捉底层解过程的渐近行为方面已被大量数值实验验证其优越性。如何从理论上解释这种优越性？本文以希尔伯特空间中的线性随机哈密顿系统为测试模型，从重对数律的角度回答了这一问题。主要贡献有两方面：首先，通过充分利用鞅的时间变换定理和Borell-TIS不等式，我们证明了精确解在特定缩放函数下的上确界几乎必然等于某个非零常数，从而确认了重对数律的有效性；其次，我们证明了随机辛全离散方法能够渐近保持重对数律，而非辛方法则不能。这揭示了随机辛方法在表征底层解过程极端波动的几乎必然渐近增长方面的优异能力。本文还将所得结果应用于线性随机振子和线性随机薛定谔方程。