Fractional Brownian flows

TL;DR

This paper investigates stochastic flows driven by fractional Brownian motion with Hurst parameter greater than 0.5, focusing on tangent flow behavior and the growth of Hausdorff measure of sub-manifolds, establishing bounds based on flow regularity.

Contribution

It provides new bounds on the growth rate of sub-manifolds under fractional Brownian flows, linking flow regularity to geometric evolution.

Findings

Bound on growth rate of Hausdorff measure

Relation between flow regularity and geometric evolution

Analysis of tangent flow behavior

Abstract

We consider stochastic flow on n-dimensional Euclidean space driven by fractional Brownian motion with Hurst parameter H greater than half, and study tangent flow and the growth of the Hausdorff measure of sub-manifolds of the ambient n-dimensional Euclidean space, as they evolve under the flow. The main result is a bound on the rate of (global) growth in terms of the (local) Holder norm of the flow.