A structure preserving stochastic perturbation of classical water wave theory

The paper proves that the free-surface variables (ζ, pφ) form a canonical pair for the stochastic classical water-wave equations by computing the variational derivatives of H and Hi and inserting them into the Stratonovich Hamilton equations (Theorem 1). It derives δH/δ(pφ) = n·x∇φ and δH/δζ = 1/2|p∇rφ|^2 − 1/2(p∂zφ)^2 + p∂zφ[p∇rφ·∇rζ] + gζ, together with δHi/δ(pφ) = n·pξi and δHi/δζ = pξi(r)·p∇rφ + p∂zφ[pξi(r)·∇rζ], using Green’s identities, the divergence theorem, the non-unit normal n = (−∇rζ, 1), and the constrained shape variation δφ = −(∂zφ)δζ on z = ζ (Proposition 3). Substituting these into the Bismut-style canonical SPDEs reproduces the stochastic kinematic boundary condition (3.8) and the stochastic Bernoulli boundary equation (4.5) exactly . The candidate solution follows the same route (same constrained variation, same boundary identities, same divergence-free ξi) and arrives at the identical functional derivatives and canonical SPDEs, hence the two arguments are essentially the same.