A Generalized Rough Super Brownian Motion

TL;DR

This paper constructs a new class of superprocesses called generalized rough super Brownian motion, derived as scaling limits of branching random walks in random environments with infinite variance offspring distributions, and analyzes their properties.

Contribution

It introduces a novel class of superprocesses with infinite variance branching, providing their construction, Laplace functional, and martingale characterization.

Findings

Laplace functional characterized by a non-linear PAM

Superprocess possesses compact support property

Exhibits exponential persistence

Abstract

In this paper, we construct scaling limits of some branching random walks in random environment whose off-spring distributions have infinite variance. The Laplace functional of the obtained random measure is given by a non-linear PAM, whose existence and uniqueness are also proved as an intermediate step. We also give a martingale characterization of above super-process and show that it possesses the compact support property and exponential persisitency.