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A regularity property of fractional Brownian sheets

Philippe Bouafia, Thierry De Pauw

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

A regularity property of fractional Brownian sheets

Abstract

A function defined on is called strongly chargeable if there is a continuous vector-field such that equals the flux of through the rectangle for all . In other words, is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter are shown to be almost surely strongly chargeable whenever
Paper Structure (10 sections, 23 theorems, 141 equations)

This paper contains 10 sections, 23 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries on $BV$ functions, $BV$ sets, and charges
  4. Strong charge functionals and strong charges
  5. The Faber-Schauder basis in $SCH([0, 1]^d)$
  6. Criteria for strong chargeability
  7. Sample paths of the Brownian sheet
  8. Hölder strong charges
  9. A Kolmogorov-type chargeability theorem for stochastic processes
  10. Sample paths of the fractional Brownian sheet

Key Result

Theorem 1

Let $U \subseteq \mathbb{R}^d$ be a bounded Lipschitz open set and $(u_n)$ be a bounded sequence in $BV(U)$. There is a subsequence $(u_{n_k})$ and a function $u \in BV(U)$ such that $u_{n_k} \to u$ in $L^1(U)$. Furthermore, $\|Du\|(U) \leqslant \liminf \|Du_{n_k}\|(U)$.

Theorems & Definitions (43)

  • Theorem : Compactness theorem, EvanGari and EvanGari
  • Theorem : Sobolev-Poincaré inequality, Ziem
  • Theorem : Trace theorem, EvanGari
  • Theorem : Extension theorem, EvanGari
  • Theorem 3.5: De Giorgi approximation
  • Proposition 3.6
  • proof
  • Theorem 4.2: Duality theorem
  • proof
  • Proposition 4.3
  • ...and 33 more