We study the fundamental problem of two gas species in two dimensional
velocity space whose molecules collide as hard circles in the presence of a
flat boundary and with dependence on only one space dimension. The case of
three-dimensional velocity space is a generalization. More speciffically the
linear problem arising when the second gas dominates as a flow with constant
velocity (and hence zero temperature) is considered. The boundary condition
adopted consists of prescribing the outgoing velocity distribution at the wall.
It is discovered that the presence of the boundary under general assumptions on
the outgoing distribution ensures the convergence of a series of path integrals
and thus a convenient representation for the solution is obtained.