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Joint continuity of the local times of fractional Brownian sheets

# Joint continuity of the local times of fractional Brownian sheets,10.1214/07-AIHP131,Annales De L Institut Henri Poincare-probabilites Et Statistiques

Joint continuity of the local times of fractional Brownian sheets
Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, 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 H\"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.
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## Citation Context (10)

• ...We observe that the power of n! in (3.15) becomes (N − βτ )/N when X is an isotropic Gaussian field as in [19] and is N − βτ when X is anisotropic in every coordinate (with the same scaling or Hölder index) as in [2]...
• ...Proof of Lemma 3.4 Even though the proof of Lemma 3.4 follows the same spirit of the proofs of Lemma 2.5 in [19] and Lemma 3.7 in [2], there are some subtle differences (see the remark above)...
• ...α2 . Inspired by Lemma 3.4 in [2], we choose...
• ...Proof of Theorem 3.3 The proof of the joint continuity of the local time of X is similar to that of Theorem 3.1 in [2] (see also the proof of Theorem 8.2 in [23])...

### Dongsheng Wu, et al. Regularity of Intersection Local Times of Fractional Brownian Motions

• ...Apropos of fractional Brownian sheets, the existence and joint continuity of the local times were discussed by Xiao and Zhang [5] as well as Ayache et al. [6]...

### Yiming Jiang, et al. Self-intersection local times and collision local times of bifractiona...

• ...(3). The family of local times Ln converges in law to a random fleld Z in C ¡ [¡D; D]d £ [0; 1]N;R ¢ as n ! 1...

### Dongsheng Wu, et al. Continuity in the Hurst index of the local times of anisotropic Gaussi...

• ...Sample path properties such as fractal dimensions and local times of fractional Brownian sheets have been studied by Ayache and Xiao [20] ,W u and Xiao [22] , Ayache, et al. [25] .M any of their results have been extended by Xiao [23] to large classes of time-anisotropic Gaussian...

### Dongsheng Wu, et al. Uniform dimension results for Gaussian random fields

• ...See, for example, Dunker [17], Mason and Shi [30], Øksendal and Zhang [31], Xiao and Zhang [41], Ayache and Xiao [6], Ayache, Wu and Xiao [5], Wu and Xiao [36] and the references therein for further information...
• ...Section 5. Our results extend the results of Ayache and Xiao [6] and Ayache, Wu and Xiao [5] for fractional Brownian sheets and Boufoussi, Dozzi and Guerbaz [13] [14] for multifractional Brownian motion to multifractional Brownian sheets...
• ...Proof The proof follows the same spirit as the proof of Lemma 2.1 of Ayache, Wu and Xiao [5]...
• ...The following lemma relates the multifractional Brownian sheet {B H(t) 0 (t)} to the independent Gaussian random fields Y‘ (‘ = 1,...,N), which is a direct extension of Lemma 2.2 of Ayache, Wu and Xiao [5]...
• ...Xiao and Zhang [41], Lemma 2.8 is proved in Ayache and Xiao [6], and Lemma 2.9 and Lemma 2.10 are from Ayache, Wu and Xiao [5]...
• ...H‘(t) > d for all t 2 I. Our results extend those of Ehm [18] for the Brownian sheet and of Ayache, Wu and Xiao [5] for fractional Brownian sheets...
• ...The main idea of proving Theorem 3.4 is similar to those in Ehm [18], Xiao [38] and Ayache, Wu and Xiao [5]...
• ...Khoshnevisan [24]]. As in Ayache, Wu and Xiao [5], the “one-sided” sectorial local nondeterministic properties of multifractional Liouville sheets proved in Section 2 [see Lemma 2.4 and Proposition 2.5] will play important rˆoles in deriving moment estimates in Lemmas 3.5 and 3.7 below...
• ...The proof of Theorem 3.4 is similar to the proofs of Theorem 4.1 in Xiao and Zhang [41] and Theorem 3.1 in Ayache, Wu and Xiao [5], and we include it here for completeness...
• ...Proof The proof is similar to the proof of Lemma 3.11 in Ayache, Wu and Xiao [5] and we include it here for completeness...

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