Variational Principles on Geometric Rough Paths and the Lévy Area Correction

The paper derives the Lévy-area-corrected Lie–Poisson generator L_s f = {h − ½ s_{ij}{Φ_i,Φ_j}, f} + ½ Σ_i {Φ_i,{Φ_i,f}} (their eq. (3.3)) and, with double-bracket dissipation (DBD), proves the invariant Gibbs density P(μ) ∝ exp(−(2ϑ/σ^2) h̃(μ)), where h̃ := h − ½ s_{ij}{Φ_i,Φ_j} (their eq. (3.9) and Theorem 3.1) by showing the Hamiltonian drift cancels and the DBD term cancels the Ito–Stratonovich correction when written as divergences of a double bracket . The candidate solution reproduces this argument: it identifies the same modified Hamiltonian h̃ via the Lévy-area correction to the generator, passes to the forward operator using divergence-free Hamiltonian flows on coadjoint orbits, and chooses β = 2ϑ/σ^2 to cancel the DBD and diffusion divergences to obtain stationarity. Aside from a minor sign slip in an intermediate sentence (corrected in the final operator), the model’s proof matches the paper’s logic and result, including the rigid-body specialization P ∝ exp(−(ϑ/σ^2) Π·I^{-1}Π + ϑ s_{ij} Π·(e_i × e_j)) which corresponds to P ∝ exp(−(2ϑ/σ^2) h̃) in so(3) .