  We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+t)
killed at the moving boundary of the cone {(t,x):0<x<t}. This article
determines the parabolic Martin boundary and all harmonic functions associated
with this process. To that end, the asymptotics of Green's functions are
determined along all directions. We also find the exit probabilities at the
edges, the probability of remaining in the cone forever and the laws of the
exit point and exit time. From this, we derive an explicit formula for the
transition kernel of the process. These results arise from two different
methods initially introduced to study random walks. An analytical approach,
developed in the 1970s by Malyshev and based on the steepest descent method on
a Riemann surface, is used to determine the asymptotics of the Green's
functions. A recursive compensation approach, inspired by the method developed
in the 1990s by Adan, Wessels and Zijm, is used to determine the harmonic
functions.
