Absolutely continuous states of exit measures for super-Brownian motions with branching restricted to a hyperplane

The exit measures of super Brownian motions with branching mechanism ψ(z)=z α, 1<α≤2, from a bounded smooth domain D in R d+1 are known to be absolutely continuous with respect to the surface area on  D if d<2α-1, whereas in the case d>1+2α-1, they are singular. However, if the branching is restricted to a singular hyperplane, it is proved that they have absolutely continuous states for all d ≥1.