Branching fractional Brownian motion: discrete approximations and maximal displacement

We construct and study branching fractional Brownian motion with Hurst parameter $H\in (1\mathrm{/}2,1)$. The construction relies on a generalization of the discrete approximation of fractional Brownian motion (Hammond and Sheffield, Probability Theory and Related Fields, 2013) to power law Pólya urns indexed by trees. We show that the first order of the speed of branching fractional Brownian motion with Hurst parameter $H$ is $c{t}^{H+1\mathrm{/}2}$ where $c$ is explicit and only depends on the Hurst parameter. A notion of “branching property” for processes with memory emerges naturally from our construction.