2412.19368
Variational integrators for stochastic Hamiltonian systems on Lie groups: properties and convergence
François Gay–Balmaz, Meng Wu
correcthigh confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves strong mean-square convergence of a Cayley-based stochastic midpoint Lie–Poisson integrator for the rigid body with truncated Wiener increments: order 1/2 for general multi-noise and order 1 when N=1. The candidate solution establishes the same rates via a different route: geometry (Casimir preservation), Lipschitz bounds on the invariant sphere, rare-event truncation estimates, and standard midpoint strong-order theory under Stratonovich commutativity. The two arguments agree on the claims and differ mainly in technique and emphasis.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper provides a principled variational derivation of a stochastic Lie–Poisson midpoint scheme and a rigorous strong mean-square convergence proof for the rigid body on SO(3). The structure-preserving properties are well established and the convergence analysis is carefully executed with appropriate stochastic inequalities. Minor clarifications would improve readability, especially around the contraction/step-size condition and the multi-noise case.