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We construct and study branching fractional Brownian motion with Hurstparameter $H\in(1/2,1)$. The construction relies on a generalization of thediscrete approximation of fractional Brownian motion (Hammond and Sheffield,Probability Theory and Related Fields, 2013) to power law P\'olya urns indexedby trees. We show that the first order of the speed of branching fractionalBrownian motion with Hurst parameter $H$ is $ct^{H+1/2}$ where $c$ is explicitand only depends on the Hurst parameter. A notion of "branching property" forprocesses with memory emerges naturally from our construction.

Branching fractional Brownian motion: discrete approximations and maximal displacement