Symplectic Runge-Kutta Methods for Hamiltonian Systems Driven by Gaussian Rough Paths
Provides theoretical foundations for symplectic integration of rough Hamiltonian systems, addressing a gap in numerical analysis for stochastic dynamics with low regularity.
This paper extends symplectic Runge-Kutta methods to Hamiltonian systems driven by Gaussian rough paths (e.g., fractional Brownian motion with Hurst parameter in (1/4,1/2]), proving that the phase flow preserves symplecticity almost surely and that the numerical methods inherit this property with pathwise convergence rates.
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 $Lip^γ$, 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.