Shi's first-passage tail conjecture for integrated stable processes

From papers

Let ZZ be a Lévy α\alpha-stable process with α(0,2]\alpha\in(0,2], and define its integrated process and first-passage time by

At=0tZsds,A_t=\int_0^t Z_s\,\mathrm{d}s, T=inf{t>0:At=1}.T=\inf\{t>0:A_t=1\}.

Shi's conjecture. Suppose that ZZ is symmetric and that α>1\alpha>1. Then

P[T>t]=t(α1)/(2α)+o(1),t.\mathbb{P}[T>t]=t^{-(\alpha-1)/(2\alpha)+o(1)},\qquad t\to\infty.

The paper says that this conjecture contradicts the authors' earlier announced result, which contained a hidden error. The source does not provide a resolution of this precise asymptotic conjecture.

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Sources & referencesView supporting material

Primary source

Thomas Simon, “On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data”, arXiv:math/0702260 (2007).

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