ON MERGING OF STOCHASTIC FLOW OF SEMI-MARKOV DYNAMICS

The paper proves eventual merging under A1–A5 by (i) deriving an exact meeting-at-next-transition formula (Theorem 4.4), (ii) ensuring eventual meeting under A4 (Theorem 4.7), (iii) computing the probability that a meeting is a merging time (Theorem 4.9), and (iv) using a product-of-failures argument to show the probability of never merging is zero (Theorem 4.12 via (4.11)–(4.12)) . The candidate solution presents a pathwise coupling with the same structural milestones: infinite jumps/no explosion from A1–A2, uniformly positive probability of meeting at the next jump under A4, a positive “stickiness” probability at meetings under A5 using the shared PRM and block structure from A3, and a geometric-tries argument to conclude a.s. merging. There is one technical slip: at meetings, the union intensity of acceptance intervals is the sum of maxima over targets, ∑k max(λik(r), λik(y+r)), not λi(r)+λi(y+r); however, the candidate only needs an upper bound, and the bound used (≤ 2∥λi∥∞) remains valid, so the overall argument still goes through. Hence both are correct, with the model’s proof differing in style and containing a minor correctable detail.