Absolute continuity of the super-Brownian motion with infinite mean

Abstract

In this work, we prove that for any dimension d≥1 and any γ∈(0,1) super-Brownian motion corresponding to the log-Laplace equation v(t,x)=(Stf)(x)−∫0t(St−svγ(s,·))(x)ds,(t,x)∈R+×Rd, is absolutely continuous with respect to Lebesgue measure at any fixed time t>0. {St}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.