The joint exit time and exit location law of a planar Ornstein-Uhlenbeck process on annular sectors and principal axis rectangles

Abstract

We study the first exit of an isotropic planar Ornstein-Uhlenbeck process from an annular sector, the region bounded by two concentric circular arcs and two radial segments, and obtain explicit eigenfunction expansions for the associated exit functionals. After the ground-state transformation that renders the generator self-adjoint, polar separation reduces the problem to a radial confluent hypergeometric equation whose two independent (Whittaker) solutions of generally non-integral order are combined through a two-radius determinant that fixes the spectrum. In this way we obtain the survival probability, the density and moments of the exit time, and -the principal contribution -the joint law of the exit time and the exit boundary, which resolves both the instant of exit and the boundary piece through which it occurs. As a companion separable case, we also treat a genuinely correlated, reversible planar Ornstein-Uhlenbeck process on a principal-axis rectangle, where the radial functions are replaced by parabolic cylinder functions: the survival probability factorizes into one-dimensional problems, while the joint law of the exit time and the exit side does not. The expansions require only one-dimensional root-finding and quadrature and are validated against Monte Carlo simulation. An application to an optically trapped colloidal particle is discussed, a setting in which the annular geometry arises naturally.