  We introduce a transient reflected Brownian motion in a multidimensional
orthant, which is either absorbed at the apex of the cone or escapes to
infinity. We address the question of computing the absorption probability, as a
function of the starting point of the process. We provide a necessary and
sufficient condition for the absorption probability to admit an exponential
product form, namely, that the determinant of the reflection matrix is zero. We
call this condition a dual skew symmetry. It recalls the famous skew symmetry
introduced by Harrison, which characterizes the exponential stationary
distributions in the recurrent case. The duality comes from that the partial
differential equation satisfied by the absorption probability is dual to the
one associated with the stationary distribution in the recurrent case.
