  We investigate the tail distribution of the first exit time of Brownian
motion with drift from a cone and find its exact asymptotics for a large class
of cones. Our results show in particular that its exponential decreasing rate
is a function of the distance between the drift and the cone, whereas the
polynomial part in the asymptotics depends on the position of the drift with
respect to the cone and its polar cone, and reflects the local geometry of the
cone at the point where the drift is orthogonally projected.
