  The distribution of exit times is computed for a Brownian particle in
spherically symmetric two- dimensional domains (disks, angular sectors, annuli)
and in rectangles that contain an exit on their boundary. The governing partial
differential equation of Helmholtz type with mixed Dirichlet- Neumann boundary
conditions is solved analytically. We propose both an exact solution relying on
a matrix inversion, and an approximate explicit solution. The approximate
solution is shown to be exact for an exit of vanishing size and to be accurate
even for large exits. For angular sectors, we also derive exact explicit
formulas for the moments of the exit time. For annuli and rectangles, the
approximate expression of the mean exit time is shown to be very accurate even
for large exits. The analysis is also extended to biased diffusion. Since the
Helmholtz equation with mixed boundary conditions is encountered in
microfluidics, heat propagation, quantum billiards, and acoustics, the
developed method can find numerous applications beyond exit processes.
