The first exit time of a Brownian motion from the Minimum and maximum parabolic domains (Q662882)

Consider a Brownian motion stating in the interior of \[ D_{\min}:= \{(x,y,z):\| x\|< \min\{(y+ 1)^{1/p}, (z+1)^{1/q}\}\} \] or \[ D_{\max}:= \{(x,y,z):\| x\|<\max\{(y+ 1)^{1/p}, (z+1)^{1/q}\}\}, \] where \(\|.\|\) denotes the Euclidean norm in \(\mathbb{R}^d\), \(y,z\geq -1\), and \(p,q> 1\). Let \(\tau_{\min}\), \(\tau_{\max}\) denote the first times from \(D_{\min}\), \(D_{\max}\), respectively. Making use of \textit{Y. Gordon's} inequality [Isr. J. Math. 50, No. 4, 265--289 (1985; Zbl 0663.60034)] and previous work of \textit{W. V. Li} [Ann. Probab. 31, No. 2, 1078--1096 (2003; Zbl 1030.60032)] and \textit{M. Lifshits} and \textit{Z. Shi} [Bernoulli 8, No. 6, 745--765 (2002; Zbl 1018.60084)] on the single parabolic domain case, the authors obtain asymptotic estimates for \(\log{\mathbf P}(\tau_{\min}> t)\) and \(\log{\mathbf P}(\tau_{\max}> t)\), \(t\to\infty\), respectively, depending on the relationship between \(p\) and \(q\).