LONG-TERM BEHAVIOUR OF ASYMPTOTICALLY AUTONOMOUS HAMILTONIAN SYSTEMS WITH MULTIPLICATIVE NOISE

The paper’s Theorem 2 is proved by a careful near-identity (energy–angle) reduction, an explicit choice of N, and a Lyapunov supermartingale V0 that absorbs the Itô contribution via a qM2 t^{-1/q} term; Doob’s inequality then yields the stability-in-probability bound. In contrast, the candidate solution relies on two unsupported steps: (i) it claims the normal-form eliminates the leading martingale term in the slow variable v so that d⟨M_ρ⟩_t = O(t^{-(n+1)/q}), which the paper neither asserts nor generally guarantees (indeed, near v→0 the diffusion coefficients scale like O(v^{-1/2}) O(t^{-1/q}), not vanishing; see Theorem 1 refinements) ; and (ii) it builds an integrating-factor supermartingale for V=ρ^2 that depends critically on that unproven quadratic-variation bound and executes a doubtful limiting argument. While the model’s conclusion matches Theorem 2, the justification conflicts with the paper’s established estimates and misses the key V0 construction and choice N=2p+q−n that make the residual term integrable .