Non-autonomous hybrid stochastic systems with delays

The statement matches Theorem 3.1 of the paper: under (3.1) and the tightness condition (3.2), (i) the pushforwards μ_t^{ρ_n}∘T_{ρ_n}^{−1} are tight on H0 and (ii) any weak limit (μ_t) is an evolution system of measures (ESM) for Z0. The paper’s proof of (i) is the standard evaluation-at-zero argument yielding (3.3), and (ii) is established by testing ϕ ∈ L_b(H0), decomposing expectations, and using (3.1) to obtain (3.5)–(3.7), which implies the evolution identity (3.4) and thus the ESM property for Z0 . The candidate solution proves (i) identically and treats (ii) via a different route: it adds and subtracts (P^0_{s,t}ϕ)∘T_{ρ_n}, uses weak convergence plus the Feller property for the ‘B_n’ term, and upgrades the uniform-in-probability convergence (3.1) to convergence of expectations uniformly on compact initial sets via a tightness–continuity lemma for the ‘A_n’ term. This alternative argument is sound and slightly stronger (it handles ϕ ∈ C_b(H0)), whereas the paper works with ϕ ∈ L_b(H0). Minor definitional tension remains because Section 2 defines ESM using C_b(H), while Section 3 verifies the identity for L_b(H0); but on Polish spaces and with Feller kernels, this is a benign issue and the proof goes through as written .