STOCHASTIC WASSERSTEIN HAMILTONIAN FLOWS

The paper’s Theorem 3.2 derives the Hamiltonian system from the dual-coordinate action S(ρ,Φ) = −∫⟨Φ,∂tρ⟩ dt + ∫H0 dt + ∫H1 dξδ, yielding ∂tρ + ∇·(ρ∇Φ) + η∇·(ρ∇Φ) ξ̇δ = 0 and ∂tΦ + 1/2|∇Φ|^2 + η(1/2)|∇Φ|^2 ξ̇δ = −δF/δρ − η δΣ/δρ ξ̇δ, with the pseudo-inverse relation (up to constants) (−Δρ)†∂tρ = (1+η ξ̇δ)Φ (the paper prints (1+ξ̇δ)Φ, which appears to be a minor omission of η). The candidate solution performs the same variational calculation and arrives at the same Hamilton equations and pseudo-inverse identity, correctly including η in the latter. However, the candidate writes the action with a minus in front of ∫H0 and then uses a sign that effectively treats it as +∫H0 when varying with respect to Φ; this small sign inconsistency does not affect the final system they present, which matches the paper’s equations and intent. Overall, both are correct in substance; the paper has a minor η-typo in the pseudo-inverse relation, and the model has a minor sign slip in the variation step, but the main result and proof approach are aligned .