Decomposition of discontinuous flows of diffeomorphisms: jumpings, geometrical and topological aspects

The paper’s Theorem 3.5 correctly proves that a Marcus SDEJ flow φt admits a local decomposition φt = Ft ∘ Gt up to a stopping time and derives the explicit Marcus equations governing Ft and Gt using a generalized Itô–Ventzel–Kunita/Leibniz formula for flows with infinitely many jumps. In contrast, the candidate solution misidentifies the SDE for the vertical factor Gt as being driven by the naive vertical projection XV of X and makes Ft depend on Gt, contradicting the paper’s nontrivial ‘tilted’ vertical component and the autonomous right-invariant equation for Ft. See the paper’s decomposition statement and equations (Theorem 3.5 and its proof) and the Leibniz formula used there .