Accepted Paper

Inserted: 29 oct 2024
Last Updated: 29 oct 2024

Journal: J. Fractal Geom.
Year: 2024

Abstract:

A function $f$ defined on $[0,1]^d$ is called strongly chargeable if there is a continuous vector-field $v$ such that $f(x_1,\cdots,x_d)$ equals the flux of $v$ through the rectangle $[0,x_1] \times \cdots \times [0,x_d]$ for all $(x_1,\cdots,x_d) \in [0,1]^d$. 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 $(H_1,\cdots,H_d)$ are shown to be almost surely strongly chargeable whenever $ \frac{H_1+\cdots+H_d}{d} > \frac{d-1}{d}. $