A regularity property of fractional Brownian sheets

Abstract

A function $f$ defined on $[0,1{]}^d$ is called strongly chargeable if there is a continuous vector field $v$ such that $f(x_1,\dots ,x_d)$ equals the flux of $v$ through the rectangle $[0,x_1]\times \cdots \times [0,x_d]$ for all $(x_1,\dots ,x_d)\in [0,1{]}^d$. In other words, $f$ is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with $d\ge 2$ parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter $(H_1,\dots ,H_d)$ are shown to be almost surely strongly chargeable whenever $H_1+\cdots +H_d>d-1$.