Occupation Time Fluctuations of Weakly Degenerate Branching Systems

We establish limit theorems for rescaled 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 nonstationary 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. (Stoch. Process. Appl. 116:1-18 and 19-35, 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. (Stoch. Process. Appl. 116:1-18 and 19-35, 2006).