Averaging for Random Metastable Systems

The paper rigorously proves quenched L1 convergence of the random invariant densities to a deterministic convex combination with weights given by the normalized averages of the opposite leak rates, under explicit hypotheses (I1)–(I6) and (P1)–(P7). Its proof crucially establishes uniform control of the density on the shrinking holes (Lemma 5.5), enabling first-order flux asymptotics of order o(ε) relative to the hole size and an explicit weight formula (Theorem 5.4). The candidate solution attains the right conclusion heuristically but omits key assumptions (notably (I5)–(I6), (P1), (P3), (P7)) and replaces the needed uniform-on-holes control by only an L1 estimate for the error gε, which is insufficient to justify the o(ε) flux expansion. It also appeals to an annealed operator to identify weights without a rigorous quenched bridge. Hence, while the final formula matches, the provided proof outline is not rigorous under the stated assumptions.