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 $d \geq 2$ 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 \[ \frac{H_1 + \cdots + H_d}{d} > \frac{d - 1}{d}. \]