Transition pathways for a class of degenerate stochastic dynamical systems with Lévy noise

The paper’s Theorem 3.1 states the Onsager–Machlup function for the degenerate SDE dX_t=g(X_t,Y_t)dt, dY_t=f(X_t,Y_t)dt + c dW_t + dL_t as OM(φ,φ̇)= 1/2 |(φ̇2−f(φ1,φ2)+∫_{|ξ|<1} ξ ν(dξ))/c|^2 + 1/2 ∂_y f(φ1,φ2), under the admissibility constraint φ̇1=g(φ1,φ2), and the tube asymptotic μ_Z(K(φ,ε)) ∝ C(ε) e^{−∫ OM dt} for small ε; this is exactly formula (3.2) with the accompanying conditions and asymptotics (3.1)–(3.3) in the paper . The proof in the paper proceeds via a Girsanov transformation to an auxiliary process, control of the degenerate component by the noisy one via Grönwall, and a path-representation/Itô calculus handling of jumps, culminating in (3.22) and the stated OM functional . The candidate solution reaches the same OM functional and admissibility condition; it conditions on the jump configuration and applies the classical diffusion OM theorem to the continuous part, arguing that jumps contribute only a φ-independent prefactor. This line is coherent with the paper’s conclusion that the jump contribution folds into the prefactor C(ε) while shifting the drift by the small-jump mean A=∫_{|ξ|<1} ξ ν(dξ) . Thus, both are correct; the paper uses a Girsanov-plus-path-representation method, while the model uses a conditioning-on-jumps reduction to a diffusion OM, i.e., different but compatible proofs.