ERRATUM: MALLIAVIN CALCULUS APPROACH TO LONG EXIT TIMES FROM AN UNSTABLE EQUILIBRIUM

In [1], an identity (37), copied here: P(ε, x) = (Equation presented). was used to derive another approximation to the quantity P(ε, x)a few lines down from (37): (2) P(ε, x) (Equation presented). The main goal of this step was to replace the density p^ε/z(u) in (1) by the Gaussian density p^z(u) in (2) with a total error o(a(ε)). However, the arguments we gave are not enough to justify this step, so let us strengthen them. We begin with an improvement of the simple uniform estimate on the difference |p^ε/z(u)− p^z(u)| given in Proposition 3.3. We will show below how one can modify its proof to obtain the following estimate: (Equation presented). improving the original claim by the factor of e^−|x−z| which takes into account the distance between x and z and can be viewed as the shifted Laplace density, up to a constant factor. We use this bound to estimate the error we make if we replace p^ε/z by p^z in (1). We take a random variable ψ independent of M_{ε, k−1} and H_{ε, k}, with a centered Laplace density fψ(x)= e^−|x|/2, x ∈ ℝ, and apply (3) to bound |p^ε/z(u)− p^z(u)| using fψ, which allows us to write (Equation presented). In the last identity, we used the independence of M_{ε, k−1} and ψ + ε^αεH_{ε, k} and the uniform boundedness of the densities of random variables M_{ε, k}, k = 1, ..., N, ε > 0. The latter is a direct consequence of Proposition 2.1.2 of [2] and the fact that the estimates (45) and (46) hold true for M_{ε, k} in place of M'_ε (this replacement requires only minimal changes in the proof). It remains to prove (3). We only need to make small changes in the proof of Proposition 3.3. Recalling the definition of u_ε from (52) of [1], we can replace the estimate (47) by (Equation presented). In the last step, besides the Cauchy-Schwarz inequality, we used that (due to Eδ(u_ε)= 0) (Equation presented). Let us show that both terms on the right-hand side of (4) decay fast enough to guarantee (3). For the first one, we recall that in the proof of the original version of Proposition 3.3, we obtained that lim_ε→0 ǁδ(u_ε)ǁ₂ = 0 uniformly in |x| < K(ε) (see the display after (55)). The other factor decays as e^−|x−z| (uniformly in x and z) due to Lemma 3.1 (or directly by the exponential martingale inequality) and the uniform boundedness of V_ε. For the second term on the right-hand side of (4), we recall that (Equation presented). is uniformly DI bounded due to (49). The other factor can be estimated as follows. Assuming z > x, for any η ∈ (0, 1), (Equation presented). We used the Cauchy-Schwarz inequality and a crude estimate of the Gaussian density of I in the third line; in the last line, we used the Markov inequality and Gaussian tails of I and M_ε (guaranteed by Lemma 3.1). Choosing η = ε^γ' for any γ' ∈ (0, 1/2) and invoking Lemma 4.2 with γ = 4γ' shows that the right-hand side of (5) is bounded by Ce^−|z−x|ε^γ'. If z < x, a similar estimate can be applied to (Equation presented). This completes the proof of (3).