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

Chuchu Chen, Xinyu Chen, Jialin Hong, Yuqian Miao

TL;DR

This work establishes a rigorous framework for the asymptotic error analysis of symplectic methods solving stochastic Hamiltonian systems (SHS) under multiplicative and additive noise. It proves stable-convergence results for the normalized error, showing the limits preserve Hamiltonian structure, and introduces a stochastic modified equation (SME) approach to derive these distributions. The paper demonstrates Hamiltonian-specific advantages of symplectic schemes for long-time simulations, supported by detailed analysis of the Kubo and linear oscillators. Numerical experiments corroborate the theoretical predictions, highlighting reduced Hamiltonian deviation growth when using symplectic integrators. Overall, the results provide both conceptual and practical insights into why symplectic methods outperform non-symplectic ones in stochastic Hamiltonian dynamics over long time horizons.

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.

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

TL;DR

This work establishes a rigorous framework for the asymptotic error analysis of symplectic methods solving stochastic Hamiltonian systems (SHS) under multiplicative and additive noise. It proves stable-convergence results for the normalized error, showing the limits preserve Hamiltonian structure, and introduces a stochastic modified equation (SME) approach to derive these distributions. The paper demonstrates Hamiltonian-specific advantages of symplectic schemes for long-time simulations, supported by detailed analysis of the Kubo and linear oscillators. Numerical experiments corroborate the theoretical predictions, highlighting reduced Hamiltonian deviation growth when using symplectic integrators. Overall, the results provide both conceptual and practical insights into why symplectic methods outperform non-symplectic ones in stochastic Hamiltonian dynamics over long time horizons.

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.

Paper Structure

This paper contains 13 sections, 16 theorems, 136 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic error distributions of symplectic methods for SHS with multiplicative noise
  4. Symplectic Euler method for the case of multiplicative noise
  5. Asymptotic error distribution of symplectic methods
  6. Hamiltonian-specific results
  7. Asymptotic error distributions of symplectic methods for SHS with additive noise
  8. A new approach for deriving the asymptotic error distribution via stochastic modified equation
  9. Construction of stochastic modified equation
  10. A new approach for deriving the asymptotic error distribution
  11. Numerical experiments
  12. Stochastic Kubo oscillator
  13. Linear stochastic oscillator

Key Result

Proposition 2.1

For each $n\in\mathbb{N}^+$, let $U_n$ be $\{\mathcal{F}_t\}$-adapted processes with sample paths in $\mathcal{D}([0,T];\mathbb{R}^d)$ and $Y$ be an $\mathbb{R}^m$-valued $\{\mathcal{F}_t\}$-semimartingale. Suppose that $(U_n,Y)\Rightarrow (U,Y)$ in $\mathcal{D}([0,T];\mathbb{R}^d\times\mathbb{R}^m) Then $\{(X_n,U_n,Y)\}$ is relatively compact and any limit point $(X,U,Y)$ satisfies If there exis

Figures (4)

  • Figure 1: $\sqrt{n}\mathbb E[H_T^n-H_T]$ and $n\mathbb E[(H_T^n-H_T)^2]$ for fixed $T$.
  • Figure 2: $n\mathbb E[(H_t^n-H_t)^2]$ for fixed $n$.
  • Figure 3: $n\mathbb E[H_T^n-H_T]$ for fixed $T$.
  • Figure 4: $n\mathbb E[H_t^n-H_t]$ for fixed $n$.

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Proposition 3.4
  • ...and 19 more