Branching fractional Brownian motion: discrete approximations and maximal displacement

TL;DR

This paper introduces a new model of branching fractional Brownian motion with Hurst parameter H in (1/2,1), providing explicit speed estimates and a natural branching property for processes with memory.

Contribution

It generalizes discrete approximations of fractional Brownian motion to branching processes with memory, establishing explicit speed and a natural branching property.

Findings

Speed of branching fractional Brownian motion is proportional to t^{H+1/2}

Construction relies on power law Pólya urns indexed by trees

Emergence of a natural branching property for processes with memory

Abstract

We construct and study branching fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. 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 P\'olya urns indexed by trees. We show that the first order of the speed of branching fractional Brownian motion with Hurst parameter $H$ is $ct^{H+1/2}$ 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.