  We consider functions which are subfunctions with respect to the differential
operator $$L_\rho = \frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} + 2\rho \frac{\partial}{\partial x} + \rho^2 $$
and are doubly periodic in the plane. These functions play an important role in
describing the asymptotic behavior of entire and subharmonic functions of
finite order. In studying their properties we are led to problems concerning
the uniqueness of Martin functions and the critical value for the parameter
$\rho$ in the homogeneous boundary problem for the operator $L_\rho$ in a
domain on the torus.
