In this work, we prove that for any dimension and any super-Brownian motion corresponding to the log-Laplace equation
is absolutely continuous with respect to Lebesgue measure at any fixed time . 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 is a finite, non-zero measure, then the log-Laplace equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.
LM was supported in part by ISF grant No. ISF 1704/18.
RM was supported in part by ISF grant No. ISF 1704/18.
"Absolute continuity of the super-Brownian motion with infinite mean." Braz. J. Probab. Stat. 35 (4) 791 - 810, November 2021. https://doi.org/10.1214/21-BJPS508