Superiority of stochastic symplectic methods via the law of iterated logarithm

TL;DR

This work theoretically explains the observed superiority of stochastic symplectic methods for SHS by establishing a law of iterated logarithm (LIL) framework. It proves the LIL for the exact linear SHS in a Hilbert space using a time-change argument and Borell–Tsirelson inequality, showing the upper growth rate scales as $\sqrt{t\log\log t}$ with a constant $\sqrt{\alpha_1^2+\alpha_2^2}\,\sup_j\sqrt{\eta_j}$. It further shows that fully discrete stochastic symplectic methods asymptotically preserve the same LIL as the exact solution, while non-symplectic schemes fail to do so, thus linking long-time almost-sure behavior to the symplectic structure. Applications to finite- and infinite-dimensional SHSs, including the linear stochastic oscillator and the linear stochastic Schrödinger equation, illustrate the concrete LIL constants and the asymptotic preservation property. These results provide a probabilistic explanation for the numerical superiority of stochastic symplectic methods in capturing the correct almost-sure long-time growth of SHS solutions, with implications for reliable long-time simulations in physics and engineering.

Abstract

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.

Theorems & Definitions (14)

• Theorem 2.1
• Proposition 2.2
• proof : Proof of Theorem \ref{['LILContinuous']}
• Proposition 2.3
• proof
• proof : Proof of Proposition \ref{['LILContinuousGeneral']}
• Theorem 3.1
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['LILfordiscrete']}