## Abstract

The mean-field stochastic partial differential equation (SPDE) corresponding to a mean-field super-Brownian motion (sBm) is obtained and studied. In this mean-field sBm, the branching-particle lifetime is allowed to depend upon the probability distribution of the sBm itself, producing an SPDE whose space-time white noise coefficient has, in addition to the typical sBm square root, an extra factor that is a function of the probability law of the density of the mean-field sBm. This novel mean-field SPDE is thus motivated by population models where things like overcrowding and isolation can affect growth. A two step approximation method is employed to show the existence for this SPDE under general conditions. Then, mild moment conditions are imposed to get uniqueness. Finally, smoothness of the SPDE solution is established under a further simplifying condition.

Originalsprog | Engelsk |
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Tidsskrift | Annals of Applied Probability |

Vol/bind | 33 |

Udgave nummer | 5 |

Sider (fra-til) | 3872-3915 |

Antal sider | 44 |

ISSN | 1050-5164 |

DOI | |

Status | Udgivet - 2023 |

### Bibliografisk note

Funding Information:Funding. Y. Hu was supported by an NSERC Discovery grant and a centennial fund from University of Alberta at Edmonton. M. A. Kouritzin was supported by an NSERC Discovery grant. J. Zheng was supported by NSFC Grant 11901598.

Publisher Copyright:

© Institute of Mathematical Statistics, 2023.