Joint continuity of the local times of fractional brownian sheets

Ayache, Antoine; Wu, Dongsheng; Xiao, Yimin

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 727-748.

Désignons par $B^H=\left\{B^H\left(t\right),t\in {ℝ}_{+}^N\right\}$ le $(N,d)$-drap Brownien fractionnaire de paramètre $H=(H_1,...,H_N)\in {(0,1)}^N$ défini par $B^H\left(t\right)=(B_1^H\left(t\right),...,B_d^H\left(t\right))(t\in {ℝ}_{+}^N)$, où $B_1^H,...,B_d^H$ sont des copies indépendantes du drap Brownien fractionnaire à valeurs réelles $B_0^H$. Nous montrons que le temps local de $B^H$ est bicontinu lorsque $d<{\sum }_{\ell =1}^N{H}_{\ell }^{-1}$. Cela résout une conjecture de Xiao et Zhang (Probab. Theory Related Fields 124 (2002)). Nous obtenons aussi des résultats fins concernant la régularité Hölderienne, locale et globale, du temps local. Ces résultats nous permettent d’étudier certaines propriétés analytiques et géométriques des trajectoires de $B^H$.

Let $B^H=\left\{B^H\left(t\right),t\in {ℝ}_{+}^N\right\}$ be an $(N,d)$-fractional brownian sheet with index $H=(H_1,...,H_N)\in {(0,1)}^N$ defined by $B^H\left(t\right)=(B_1^H\left(t\right),...,B_d^H\left(t\right))(t\in {ℝ}_{+}^N)$, where $B_1^H,...,B_d^H$ are independent copies of a real-valued fractional brownian sheet $B_0^H$. We prove that if $d<{\sum }_{\ell =1}^N{H}_{\ell }^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

Classification : 60G15, 60G17
Mots clés : fractional brownian sheet, Liouville fractional brownian sheet, fractional brownian motion, sectorial local nondeterminism, local times, joint continuity, Hölder conditions

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