Absolute continuity of the super-Brownian motion with infinite mean

In this work, we prove that for any dimension d ≥ 1andanyγ ∈ (0, 1) super-Brownian motion corresponding to the log-Laplace equation (Formula Presented) is absolutely continuous with respect to Lebesgue measure at any fixed time t>0. {S_t }_t≥0 denotes a transition semigroup of a standard Brownian motion. Our proof is based on properties of solutions of the log-Laplace equation. We also prove that when initial datum v(0, ·) is a finite, non-zero measure, then the log-Laplace equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.