Synchronization and random attractors for reaction jump processes

The paper proves that the embedded birth–death RDS has a two-point random attractor that is a random period‑2 orbit, is both pullback and forward attracting, and whose two points are adjacent; it does so via two‑point motion hitting the thick diagonal and general attractor theory. The candidate solution establishes the same conclusions with a different route (two‑step monotonicity on parity classes, monotone pullback limits plus tightness, an expectation argument for adjacency, and a synchronous-coupling/Foster–Lyapunov coalescence proof). Aside from a minor sign oversight in the interval length in Step 4 and a place where independence at stopping times should be stated explicitly in Step 5, the candidate solution’s logic aligns with the paper’s results.