We consider the full weak convergence, in appropriate function spaces, of systems of noninteracting particles undergoing critical branching and following a self-similar spatial motion with stationary increments. The limit processes are measure-valued, and are of the super and historical process type. In the case in which the underlying motion is that of a fractional Brownian motion, we obtain a characterization of the limit process as a kind of stochastic integral against the historical process of a Brownian motion defined on the full real line.
Robert J. Adler. Gennady Samorodnitsky. "Super Fractional Brownian Motion, Fractional Super Brownian Motion and Related Self-Similar (Super) Processes." Ann. Probab. 23 (2) 743 - 766, April, 1995. https://doi.org/10.1214/aop/1176988287