The first exit time of Brownian motion from a parabolic domain

Abstract

Consider a planar Brownian motion starting at an interior point of the parabolic domain $D=\{ (x,y): \; y>x^2\}$, and let $\tau_D$ denote the first time the Brownian motion exits from $D$. The tail behaviour (or equivalently, the integrability property) of $\tau_D$ is somewhat exotic since it arises from an interference of large-deviation and small-deviation events. Our main result implies that the limit of $T^{-1/3} \log\mathbb{P}\{ \tau_D >T\}$, $T\to \infty$, exists and equals $-3\pi^2/8$, thus improving previous estimates by Bañuelos et al. and Li. The existence of the limit is proved by applying the classical Schilder large-deviation theorem. The identification of the limit leads to a variational problem, which is solved by exploiting a theorem of Biane and Yor relating different additive functionals of Bessel processes. Our result actually applies to more general parabolic domains in any (finite) dimension.