Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty

Both the paper and the candidate correctly establish local pathwise strong well-posedness with a blow-up alternative and pathwise uniqueness; the main statements (Theorem 5 and the maximality property) match in content and scope. However, for the global weak theory at α=β=−1/2, the paper states a sharp energy inequality with unit constant (its (17)), but it does not address the non-vanishing gravity coupling g⟨u,∇h⟩ needed to justify that exact bound unless g=0 or an auxiliary constraint (e.g. zero-mean h with Poincaré) is imposed. The candidate solution explicitly handles this term, giving the unit-constant inequality in the g=0 case and a Gronwall-type bound for general g. Hence the model’s solution is more precise about assumptions needed for (17), while the paper leaves them implicit. The local strong theory (construction via truncation/projection, commutator controls, noise cancellations, uniqueness, and maximality) is sound and consistent with the candidate’s outline.