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For a group hyperbolic relative to virtually nilpotent subgroups, on a cuspedgraph associated to the group, we construct a random walk whose Martin boundaryis the Bowditch boundary of the group. Moreover, the harmonic measure is aconformal density corresponding to a hyperbolic Green metric and is exactdimensional on the Bowditch boundary. The latter equipped with a visualdistance induced by the Green metric is an Ahlfors-regular metric measurespace. The dimension is given in terms of the drift, a Green drift and theasymptotic entropy. The Patterson-Sullivan density for the action on the cuspedgraph in this case is doubling, its dimension is obtained by looking at cuspexcursions of geodesics.

Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups