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

Yuri Bakhtin, Hong Bin Chen, Zsolt Pajor-Gyulai

Research output: Contribution to journalArticlepeer-review

## Abstract

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 pε/z(u) in (1) by the Gaussian density pz(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 |pε/z(u)− pz(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 pε/z by pz in (1). We take a random variable ψ independent of Mε, k−1 and Hε, k, with a centered Laplace density fψ(x)= e−|x|/2, x ∈ ℝ, and apply (3) to bound |pε/z(u)− pz(u)| using fψ, which allows us to write (Equation presented). In the last identity, we used the independence of Mε, k−1 and ψ + εαεHε, k and the uniform boundedness of the densities of random variables Mε, 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 Mε, 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 uε 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(uε)= 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 limε→0 ǁδ(uε2 = 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 Vε. 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 Mε (guaranteed by Lemma 3.1). Choosing η = εγ' 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|εγ'. If z < x, a similar estimate can be applied to (Equation presented). This completes the proof of (3).

Original language English (US) 3199-3200 2 Annals of Applied Probability 32 4 https://doi.org/10.1214/21-AAP1751 Published - Aug 2022

## Keywords

• Exit problem
• Malliavin calculus
• polynomial decay
• unstable equilibrium
• vanishing noise limit

## ASJC Scopus subject areas

• Statistics and Probability
• Statistics, Probability and Uncertainty

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