The first exit time of Brownian motion form a parabolic domain
Let \(\{B_t,t\geq 0\}\) be a Brownian motion taking values in \(R^{d+1}\), \(d\geq 1\), starting at a point in the interior of the generalized parabolic shape \(D=D_{d,p,a}=\{(x,y)=(x_1,\dots,x_d,y)\in R^{d+1}: y\geq \|x\|^p\}\), where \(p>1\) and \(\|x\|\) is the Euclidean norm of \(x\) in \(R^{d}\). Denote by \(\tau_D\) the first exit time of \(B(t)\) from \(D\). The authors focus on asymptotic behaviour of \(P(\tau_D>T)\) as \(t\to \infty\) and obtain an explicit value of \(\lim_{t\to \infty}T^{-(p-1)/(p+1)}\log P(\tau_D>T)\). The existence of the limit is proved by using the large deviation technique, namely Shilder theorem. The identification of the limit leads to a variational problem, which is solved by exploiting a theorem of \textit{P. Biane} and \textit{M. Yor} [Bull. Sci. Math., II. Sér. 111, 23-101 (1987; Zbl 0619.60072)] relating additive functionals of Bessel process. For \(d=a=1\) and \(p=2\) the main result immediately implies that \(\lim_{t\to \infty}T^{-1/3}\log P(\tau_D>T)=-3\pi^2/8\). The obtained results are extended to more general domains with regularly varying boundary in any (finite) dimension. The case of non-Euclidean norms is also discussed and open problems are formulated.
scientific article; zbMATH DE number 1877109