# Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths

## TL;DR

This paper develops symplectic Runge-Kutta methods for Hamiltonian systems driven by Gaussian rough paths, demonstrating their structure-preserving properties and convergence, with numerical validation.

## Contribution

It introduces symplectic Runge-Kutta methods for rough path driven Hamiltonian systems, establishing their solvability, convergence rates, and structure preservation.

## Key findings

- Phase flow preserves symplectic structure almost surely.
- Symplectic Runge-Kutta methods inherit symplecticity.
- Numerical experiments confirm theoretical results.

## Abstract

We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter $H\in(1/4,1/2]$. We indicate that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge--Kutta methods, which are implicit methods in general. If the vector fields belong to $Lip^{\gamma}$, we obtain the solvability of Runge--Kutta methods and the pathwise convergence rates. For linear and skew symmetric vector fields, we focus on the midpoint scheme to give corresponding results. Numerical experiments verify our theoretical analysis.