On mean-field super-Brownian motions

TL;DR

This paper introduces a new mean-field super-Brownian motion model governed by a novel SPDE that incorporates distribution-dependent branching effects, with proofs of existence, uniqueness, and smoothness of solutions.

Contribution

It develops a new mean-field SPDE for super-Brownian motions with distribution-dependent branching, including existence, uniqueness, and smoothness results.

Findings

Existence of solutions under general conditions

Uniqueness under mild moment conditions

Smoothness of solutions under additional assumptions

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 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.