Symplectic methods for stochastic Hamiltonian systems: asymptotic error distributions and Hamiltonian-specific analysis

2025/12/03 by Chuchu Chen, Chen, Chuchu, Xinyu Chen +5
Decision Sciences · Mathematics · Physics and Astronomy · #60F05 #60H35 #65P10 #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2512.03840

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems and further provide Hamiltonian-specific analysis that clarifies the superiority of symplectic methods. Our contribution is threefold. First, we derive the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems with multiplicative noise and additive noise, respectively, and show that the obtained limiting stochastic processes satisfy equations retaining the Hamiltonian formulations. Second, we propose a new approach for calculating the asymptotic error distribution, revealing the connection between the stochastic modified equation and the asymptotic error distribution. Third, we characterize the limiting distribution of the normalized Hamiltonian deviation, thereby illustrating through test equations the superiority of symplectic methods for long-time simulations of the Hamiltonians, even in the limit as the step size tends to zero.

Citations

• Asymptotic error distribution for stochastic Runge--Kutta methods of strong order one
• Asymptotic error distribution of accelerated exponential Euler method for parabolic SPDEs
• Superiority of stochastic symplectic methods via the law of iterated logarithm
• Error Distribution for One-Dimensional Stochastic Differential Equation Driven By Fractional Brownian Motion
• Backward Euler method for stochastic differential equations with non-Lipschitz coefficients
• Error Distribution Of The Euler Approximation Scheme For Stochastic Volterra Integral Equations
• Large deviations principles for symplectic discretizations of stochastic linear Schrödinger Equation
• The probabilistic superiority of stochastic symplectic methods via large deviations principles
• Computing effective diffusivity of chaotic and stochastic flows using structure preserving schemes
• Asymptotic error distribution for the Euler scheme with locally\n Lipschitz coefficients
• Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Related

• Superiority Of Symplectic Methods For Stochastic Hamiltonian System Via Asymptotic Error Distribution
• The probabilistic superiority of stochastic symplectic methods via large deviations principles
• Modified equations for weak stochastic symplectic schemes via their generating functions
• Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation
• Symplectic methods based on Pad\acutee approximation for some stochastic Hamiltonian systems