Occupation Time Fluctuations of Weakly Degenerate Branching Systems

TL;DR

This paper proves limit theorems for occupation time fluctuations in anisotropic, weakly degenerate branching particle systems, revealing new Gaussian fields and complex temporal structures across different dimensions.

Contribution

It introduces new limit processes for occupation time fluctuations in weakly degenerate systems with anisotropic motion, extending previous models.

Findings

Large dimension limits yield operator-scaling Gaussian fields with non-stationary increments.

Intermediate and critical dimensions show complex spatial structures similar to non-degenerate cases.

Temporal structures differ significantly from non-degenerate models due to weak degeneracy.

Abstract

We establish limit theorems for re-scaled occupation time fluctuations of a sequence of branching particle systems in $\R^d$ with anisotropic space motion and weakly degenerate splitting ability. In the case of large dimensions, our limit processes lead to a new class of operator-scaling Gaussian random fields with non-stationary increments. In the intermediate and critical dimensions, the limit processes have spatial structures analogous to (but more complicated than) those arising from the critical branching particle system without degeneration considered by Bojdecki et al.{\it Stochastic Process. Appl. 2006}. Due to the weakly degenerate branching ability, temporal structures of the limit processes in all three cases are different from those obtained by Bojdecki et al. .{\it Stochastic Process. Appl. 2006}.