## Abstract

Consider a planar Brownian motion starting at an interior point of the parabolic domain D = {(x, y) : y > x^{2}}, and let τ_{D} denote the first time the Brownian motion exits from D. The tail behaviour (or equivalently, the integrability property) of τ_{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 ℙ{τ_{D} > T}, T → ∞, exists and equals -3π^{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.

Original language | English |
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Pages (from-to) | 745-765 |

Number of pages | 21 |

Journal | Bernoulli |

Volume | 8 |

Issue number | 6 |

Publication status | Published - 1 Dec 2002 |

## Scopus subject areas

- Statistics and Probability