Lyapunov Densities For Markov Processes: An Application To Quantum Systems With Non-Demolition Measurements

The paper’s Theorem 2 states that if a Markov operator P admits a properly subinvariant, positive function u that is locally integrable with respect to an admissible family A, then P is sweeping with respect to A. The proof in the paper proceeds by chaining known results: (i) proper subinvariance implies dissipativity via a Foguel lemma, (ii) existence of a subinvariant function yields smoothing, and (iii) dissipativity plus smoothing implies sweeping; see the definitions and statements in the paper (sweeping, admissible families, smoothing; Lemma 2; Lemma 4; Theorem 1; Theorem 2) . The candidate solution instead gives a direct potential-theoretic argument via the telescoping identity u − P^n u = ∑_{k=0}^{n−1} P^k(u−Pu), controls the localized series using the local integrability of u, and then transfers decay from g := u−Pu to arbitrary f ∈ L1 by approximation and L1-contraction. Both arguments are logically sound given the same assumptions. The approaches are different: the paper uses an abstract chain of ergodic-theory lemmas; the model uses a constructive telescoping/dominating argument.