Correction to: Mean exit time for the overdamped Langevin process: the case with critical points on the boundary (Communications in Partial Differential Equations, (2021), 46, 9, (1789-1829), 10.1080/03605302.2021.1897841)

Abstract

0 such that for h small enough: (Formula presented.) Before proving (1), we give some preliminary estimates. By [1, Proposition 14], for any (Formula presented.) it holds for h small enough: (Formula presented.) We now claim that there exist (Formula presented.) and c > 0 such that for h small enough: (Formula presented.) and for all bounded and measurable function (Formula presented.) (Formula presented.) Equation (3) is a consequence of [2, Theorem 1] which implies that (recall (Formula presented.)): (Formula presented.) Let us prove (4). Set (Formula presented.) We have for all (Formula presented.) (Formula presented.) In addition, with the same arguments as used to prove [1, Eq. (27)], there exist c > 0 and (Formula presented.) such that for all h > 0 small enough and (Formula presented.) Consequently, it holds: (Formula presented.) In addition, with the same arguments as those used to prove [1, Lemma 3] (which rely on a Schauder estimate, see also [3, Lemma 1]), we have (Formula presented.) Then, (4) holds with (Formula presented.) replaced by (Formula presented.) From this, we deduce that for all (Formula presented.) using the strong Markov property, (Formula presented.) This ends the proof of (4). We now show (1). By the same arguments as those used to prove [4, Lemma 6] and using (4), we deduce for all (Formula presented.) (Formula presented.) where c > 0 is independent of h and (Formula presented.) Then, (2), (3), and [1, Proposition 5] (with (Formula presented.) there) imply (1). Step 2: we now end the proof of [1, Theorems 1 and 2]. Let (Formula presented.) We have by the strong Markov property, for all (Formula presented.) (Formula presented.) Assume that (Formula presented.) Recall that in this case, (Formula presented.) (see [1, Eq. (8)]). Then, using (6), [1, Lemma 3], [1, Proposition 5] (with (Formula presented.) there), [1, Eq. (65)], and (1), we deduce that: (Formula presented.) This previous estimates extend uniformly to any (Formula presented.) using the same arguments as those used in Step 4 of the proof of [1, Proposition 5]. Let us now consider (Formula presented.) By (6), [1, Lemma 3], [1, Proposition 5], and [1, Eq. (66)], we then have for all (Formula presented.) (Formula presented.) uniformly in (Formula presented.) Finally, by (1, 6), and the lines just after [1, Eq. (66)], there exists (Formula presented.) such that if (Formula presented.) the previous inequality is an equality which holds uniformly in (Formula presented.) (and is then extended uniformly to any (Formula presented.) using the same arguments as those used in Step 4 of the proof of [1, Proposition 5]). This concludes the proof of [1, Theorem 2], and the proofs of items 2 and 3 in [1, Theorem 1]. The proof of [1, Theorem 1] is thus complete. □ Remark. We mention that with this erratum, [1, Proposition 6] is now not needed in this work.