We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension d≥3, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in d≥4. In d=3, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for d=3 also a functional limit theorem.
"A functional CLT for the occupation time of a state-dependent branching random walk." Ann. Probab. 35 (6) 2063 - 2090, November 2007. https://doi.org/10.1214/009117907000000150