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November 2021 Absolute continuity of the super-Brownian motion with infinite mean
Rustam Mamin, Leonid Mytnik
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Braz. J. Probab. Stat. 35(4): 791-810 (November 2021). DOI: 10.1214/21-BJPS508

Abstract

In this work, we prove that for any dimension d1 and any γ(0,1) super-Brownian motion corresponding to the log-Laplace equation

v(t,x)=(Stf)(x)0t(Stsvγ(s,·))(x)ds,(t,x)R+×Rd,

is absolutely continuous with respect to Lebesgue measure at any fixed time t>0. {St}t0 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.

Funding Statement

LM was supported in part by ISF grant No. ISF 1704/18.
RM was supported in part by ISF grant No. ISF 1704/18.

Citation

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Rustam Mamin. Leonid Mytnik. "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

Information

Received: 1 December 2020; Accepted: 1 May 2021; Published: November 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/21-BJPS508

Keywords: stable branching , Superprocesses

Rights: Copyright © 2021 Brazilian Statistical Association

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Vol.35 • No. 4 • November 2021
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