Some remarks on stochastic converse Lyapunov theorems

The paper proves a tractable stochastic converse Lyapunov theorem by converting an ω-uniform KL trajectory bound into a path-sum Lyapunov function L and establishes a one-step high-probability negative drift without needing one-step contractivity; the key step is the representation β(v,t)=κ1(κ2(v)e^{-t}) (Proposition 7 in [19]) and the construction L′(trajectory)=∑_{t≥0} c′(d_{G′}(Z_t)), with c′=κ_1^{-1}, followed by L(s)=ess sup_ω L′[trajectory] on a high-probability event, yielding i)–ii) in Theorem 6 . The model instead proposes a series-based telescoping construction that requires an unproven eventual one-step contractivity of α_β(r)=β(r+C_0,1); it explicitly acknowledges this gap and only obtains a result for a restrictive subclass (e.g., separable exponentials), leaving the general weakly tractable case unresolved. Hence the model’s solution is incomplete. The paper’s main drift argument and K∞ sandwiching follow from its KL-to-exponential form and path-sum construction , but one technical point—local Lipschitz regularity of L—rests on an unsubstantiated claim that the series’ summands are Lipschitz in the initial state without any continuity assumption on the Markov kernel, so the local-Lipschitz part of Theorem 6 appears under-justified, although likely repairable with mild additional hypotheses or a smoothing step. Therefore, both are incomplete: the model on substance (missing a key step), and the paper on a technical regularity point.