  For a group hyperbolic relative to virtually nilpotent subgroups, on a cusped
graph associated to the group, we construct a random walk whose Martin boundary
is the Bowditch boundary of the group. Moreover, the harmonic measure is a
conformal density corresponding to a hyperbolic Green metric and is exact
dimensional on the Bowditch boundary. The latter equipped with a visual
distance induced by the Green metric is an Ahlfors-regular metric measure
space. The dimension is given in terms of the drift, a Green drift and the
asymptotic entropy. The Patterson-Sullivan density for the action on the cusped
graph in this case is doubling, its dimension is obtained by looking at cusp
excursions of geodesics.
