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Mean-Square Convergence of a New Parameterized Leapfrog Scheme for Hamiltonian Systems Driven by Gaussian Process Potentials

Sourabh Bhattacharya

TL;DR

This work analyzes a parameterized stochastic leapfrog integrator for Hamiltonian systems with Gaussian process potentials, proving a mean-square convergence rate of $O(\Delta t)$ under minimal regularity. By conducting pathwise expansions, deriving a modified equation, and establishing a sharp local truncation error bound, the authors show that the global mean-square error remains controlled over finite time horizons. A key finding is that choosing $\alpha_1=\beta_1=0$ aligns the modified dynamics with the original Hamiltonian flow at leading order while enabling numerical stabilization through higher-order terms. The results provide a rigorous foundation for using a GP-driven, nonparametric potential within a symplectic leapfrog framework, with potential applications in Bayesian sampling, molecular dynamics, and stochastic geometric mechanics where GP potentials arise.

Abstract

This paper establishes the mean-square convergence of a new stochastic, parameterized leapfrog scheme introduced in our companion paper Mazumder et al. (2026) for Hamiltonian systems with Gaussian process potentials. We consider a one-step numerical integrator and provide a complete, rigorous analysis under minimal regularity assumptions on the Gaussian potential. The key technical contribution is identifying and exploiting the symplectic structure ingrained in our stochastic, parameterized leapfrog method. Combined with local truncation error analysis, this leads to a global error bound of O(δt) in mean-square sense. Our results establish that although the spatio-temporal model of Mazumder et al. (2026) arises as the anticipated new stochastic leapfrog solution of a system of modified (parameterized) stochastic Hamiltonian equations, the new stochastic leapfrog actually solves the traditional stochastic Hamiltonian equations, driven by Gaussian process potential.

Mean-Square Convergence of a New Parameterized Leapfrog Scheme for Hamiltonian Systems Driven by Gaussian Process Potentials

TL;DR

This work analyzes a parameterized stochastic leapfrog integrator for Hamiltonian systems with Gaussian process potentials, proving a mean-square convergence rate of under minimal regularity. By conducting pathwise expansions, deriving a modified equation, and establishing a sharp local truncation error bound, the authors show that the global mean-square error remains controlled over finite time horizons. A key finding is that choosing aligns the modified dynamics with the original Hamiltonian flow at leading order while enabling numerical stabilization through higher-order terms. The results provide a rigorous foundation for using a GP-driven, nonparametric potential within a symplectic leapfrog framework, with potential applications in Bayesian sampling, molecular dynamics, and stochastic geometric mechanics where GP potentials arise.

Abstract

This paper establishes the mean-square convergence of a new stochastic, parameterized leapfrog scheme introduced in our companion paper Mazumder et al. (2026) for Hamiltonian systems with Gaussian process potentials. We consider a one-step numerical integrator and provide a complete, rigorous analysis under minimal regularity assumptions on the Gaussian potential. The key technical contribution is identifying and exploiting the symplectic structure ingrained in our stochastic, parameterized leapfrog method. Combined with local truncation error analysis, this leads to a global error bound of O(δt) in mean-square sense. Our results establish that although the spatio-temporal model of Mazumder et al. (2026) arises as the anticipated new stochastic leapfrog solution of a system of modified (parameterized) stochastic Hamiltonian equations, the new stochastic leapfrog actually solves the traditional stochastic Hamiltonian equations, driven by Gaussian process potential.
Paper Structure (29 sections, 7 theorems, 97 equations)

This paper contains 29 sections, 7 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Original Hamiltonian system with Gaussian process potential
  4. The stoachstic parameterized leapfrog scheme from main
  5. Notation and basic definitions
  6. Gaussian process framework from main
  7. Assumptions
  8. Moment bounds for Gaussian process derivatives
  9. Pathwise analysis of the new leapfrog scheme
  10. Local truncation error analysis for our stochastic leapfrog scheme
  11. Main convergence theorem for our leapfrog scheme
  12. Conclusion
  13. Complete proofs
  14. Proof of Theorem \ref{['thm:moment_bounds']}
  15. Notation and preliminaries
  16. ...and 14 more sections

Key Result

Theorem 2.4

Under Assumption ass:gp_regularity, for any compact set $D \subset \mathbb{R}^d$, there exist finite constants $C_1(D), C_2(D), C_3(D) > 0$ such that:

Theorems & Definitions (25)

  • Definition 2.1: Gaussian process
  • Remark 2.3: Sample path regularity
  • Theorem 2.4: Moment bounds for derivatives
  • Remark 2.5
  • Remark 2.7: Discussion of Assumption \ref{['ass:mass_matrix']}
  • Remark 2.9: Discussion of Assumption \ref{['ass:param_expansions']}
  • Remark 2.11: Justification of Assumption \ref{['ass:bounded_iterates']}
  • Lemma 3.1: Pathwise Expansions of the Modified Scheme
  • Remark 3.2: Discussion of Lemma \ref{['lem:pathwise_expansions']}
  • Theorem 3.3: Modified stochastic ODE corresponding to our leapfrog scheme
  • ...and 15 more