The first exit time of Brownian motion from a parabolic domain

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 To 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 P {tau(D) > T}, T --> infinity, exists and equals -3pi(2)/8, thus improving previous estimates by Banuelos 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.