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

Sourabh Bhattacharya|arXiv (Cornell University)|Feb 15, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper proves mean-square convergence of a parameterized stochastic leapfrog scheme for Hamiltonian systems with Gaussian process potentials, achieving first-order mean-square global accuracy under specific parameter assumptions.

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.

Motivation & Objective

• Motivate numerical integration of Hamiltonian systems with Gaussian process potentials.
• Introduce and formalize the parameterized stochastic leapfrog scheme.
• Establish mean-square convergence and provide a global error bound.
• Derive a modified ODE description and local truncation error analysis under minimal regularity.
• Outline proofs under precise Gaussian process assumptions and discuss implications for nonparametric potentials.

Proposed method

• Define the original Hamiltonian system with Gaussian process potential and the parameterized leapfrog scheme (y_{n+1}, x_{n+1}) with parameters alpha and beta.
• Perform pathwise (sample-path) expansions to reveal one-step updates and parameter effects.
• Derive a modified stochastic ODE that the scheme tracks up to O(delta t^2) accuracy.
• Carry out a local truncation error analysis to bound the one-step discrepancy in mean-square sense.
• Prove a main convergence theorem showing mean-square convergence of order 1 over finite time horizons.
• Use Gaussian process moment bounds and Borell–TIS inequality to control derivatives of V and ensure finite moments.

Experimental results

Research questions

• RQ1 Does the parameterized leapfrog scheme converge in mean-square to the exact Hamiltonian flow with Gaussian process potential?
• RQ2 What is the order of mean-square convergence and under what parameter choices is it achieved?
• RQ3 How do the Gaussian process derivative bounds influence stability and error growth?
• RQ4 How do the introduced parameters alpha and beta affect the modified equation and convergence rate?

Key findings

• A global mean-square convergence order of 1 is established for the modified leapfrog scheme over finite intervals.
• Under alpha1 = beta1 = 0, the local truncation error is O(delta t^4) in mean-square, aligning with an optimal 1st-order method.
• A modified continuous-time ODE characterizes the scheme, revealing parameter-driven corrections to the standard Hamiltonian dynamics.
• Moment bounds for Gaussian process derivatives are derived, ensuring finite second moments essential for the analysis.
• The analysis shows the scheme actually solves the traditional stochastic Hamiltonian equations driven by Gaussian process potential in the mean-square sense.