Erratum to: Asymptotic Estimates of the Distribution of Brownian Hitting Time of a Disc

Erratum to: J Theor Probab (2012) 25:450–463 DOI 10.1007/s10959-010-0305-8

In my paper [1] it is purported stated that Theorem 1 immediately follow from Lemmas 4 and 5, which statement is incorrect (at least for x fixed the error estimate of Theorem 1 does not follow from Lemmas 4 and 5). For it to be correct, the following slight modification of Lemma 4 is sufficient.

FormalPara Lemma 4′

Uniformly for |x|>r ₀, as t→∞

$$p_{r_0, x}(t)-q_x(t) =O \biggl(\frac{1+ \lg^+ |x|}{t^2(\lg t)^2}\wedge \frac{1}{|x|^4 \bigl(1+ \lg^+ |x|\bigr)} \biggr) $$

and the difference $p_{r_{0},x}(t)-q^{c}_{x}(t)$ admits the same estimate.

If t ^δ<|x|<t ^1/2 for some δ>0, this is the same as Lemma 4 of [1]. Hence for the proof of Lemma 4′ we can suppose that |x|≤t ^1/4 and it suffices to prove

$$ \frac{1}{2\pi}\int_{-\infty}^\infty \frac {2iu}{g(iu)}w(u)K_0(|x|\sqrt{2iu}\,)e^{itu}\,du=O \biggl( \frac{1+ \lg^+ |x|}{t^2(\lg t)^2} \biggr) $$

(1)

in place of Eq. (16) (in [1]). (Here $g(z)= -\lg(2^{-1}e^{\gamma }r_{0}\sqrt{2z}\,)$ and w is a smooth function that equals 1 in a neighborhood of the origin and vanishes outside a finite interval.) Let r ₀=1 for simplicity. The leading term of $K_{0}(|x|\sqrt {2iu}\,)$ is g(ix ² u)=−lg|x|+g(iu) and its contribution to the above integral equals

where N may be any positive constant and the last equality may be derived by using Eq. (35), which implies $\int_{-\infty}^{\infty}(\lg iu)^{-1}e^{itu}\,du = [2\pi/t(\lg t)^{2}](1+o(1))$.

Put $h(z) = K_{0}(\sqrt{2z})-g(z)$ and V(u)=h(i|x|² u). We must obtain a uniform bound for the integral I:=∫[w(u)uV(u)/g(iu) ] e ^itu  du. Noting that h(z)=O(zlgz),h′(z)=O(lgz) and h ^(j)(z)=O(z ^−j+1) (j=2,3,4) for z=iy with y∈R∖{0}, we have

(2)

We can integrate by parts twice the integral that defines I, transforming it into

Then we split the range of this integral at |u|=1/t and apply to the integral on |u|>1/t integration by parts twice more, which with the help of (2) gives for it the bound O(x ²/t) valid uniformly for $|x|\leq\sqrt{t}$ (cf. [2], Lemma 2.2), so that I=O(x ²/t ³), a bound sufficient for the required estimate. This completes the proof of Lemma 4′.

In addition there are simple errors on p. 456: in the second formula on the fifth line from the bottom of the page the bound O(y∧1) must be replaced by O(lg(1+y)) and also in the next line $O(e^{-2\sqrt{y}})$ by O(y ^−1/4). These errors only require a few simple modifications to the arguments (tacitly) involved there.