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

TL;DR

This paper proves the mean-square convergence of a novel stochastic leapfrog integrator for Hamiltonian systems with Gaussian process potentials, demonstrating its accuracy and structure-preserving properties under minimal assumptions.

Contribution

It introduces and rigorously analyzes a new parameterized leapfrog scheme that converges in mean-square sense for stochastic Hamiltonian systems driven by Gaussian processes.

Findings

Global error bound of O(Δt) in mean-square sense.

The scheme preserves the symplectic structure.

It solves traditional stochastic Hamiltonian equations driven by Gaussian potentials.

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({\delta}t) 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.