The Compact Support Property of Rough Super Brownian Motion on $\mathbb{R}^2$

TL;DR

This paper proves that rough super-Brownian motion in two-dimensional space has the compact support property, using interior estimation techniques to overcome challenges posed by the singular SPDE nature of the process.

Contribution

It establishes the compact support property for rough super-Brownian motion via novel interior estimation methods, despite the singularities of the associated SPDE.

Findings

Compact support property holds for rough super-Brownian motion in b2.

Interior estimation method effectively handles singular SPDE challenges.

Supports the understanding of measure-valued processes in random environments.

Abstract

We discuss the compact support property of the rough super-Brownian motion constructed as a scaling limit of a branching random walk in static random environment. The semi-linear equation corresponding to this measure-valued process is the continuous parabolic Anderson model, a singular SPDE in need of renormalization, which prevents the use of classical PDE arguments. But with the help of an interior estimation method, we are able to show that the compact support property also holds for rough super-Brownian motion.