Resonance in Isochronous Systems with Decaying Oscillatory and Stochastic Perturbations

Theorem 2 in the paper states exactly the stochastic stability-in-probability bound the candidate addresses, with the same scaling ML, the same horizon T depending on integrability of μ^{2p−q+n}, and the same assumptions p > (q−n)/2 and Re β01 < 0, Re β̃02 < 0 (definition of β̃02 is given before Theorem 2). The paper’s proof constructs a tailored quadratic Lyapunov function U0 = V0 with a cross-term chosen to neutralize linear off-diagonal couplings, and then shows LU0 ≤ −C μ^q |z|^2 + ε^2 C1 μ^{2p−q+n}; it then makes V := U0 + ε^2 U1(t,T) a supermartingale and applies the standard stopping-time/Doob bound to obtain (20) with the stated choices of T (including Tε when μ^{2p−q+n} ∉ L^1) . By contrast, the candidate uses the Euclidean Lyapunov U(z)=|z|^2 and claims to absorb off-diagonal terms directly into the diagonal damping. That absorption is valid when q>n (since μ^{(q−n)/2}→0), but in the boundary case q=n the off-diagonal terms are the same order as the diagonal ones, so the Euclidean quadratic form need not yield a negative symmetric part even if the linearization has stable eigenvalues. The paper resolves this with an appropriate quadratic form V0 carrying a cross-term; the candidate does not. The rest of the candidate’s steps (generator inequality structure, stopping-time/Dynkin or Doob, and the choice of T via γ_{2p−q+n}) match the paper’s proof strategy and rates, but the crucial linear-drift negativity is unjustified in general with U=|z|^2. Hence the paper is correct and the model solution is incomplete/wrong on a key step.