Let B H = B H (t), t ∈ R N + be an (N, d)-fractional Brownian sheet with Hurst index H = (H 1 , . . . , H N ) ∈ (0, 1) N . Our objective of the present article is to characterize the anisotropic nature of B H in terms of H . We prove the following results:(1) B H is sectorially locally nondeterministic.(2) By introducing a notion of "dimension" for Borel measures and sets, which is suitable for describing the anisotropic nature of B H , we determine dim H B H (E) for an arbitrary Borel set E ⊂ (0, ∞) N . Moreover, when B α is an (N, d)-fractional Brownian sheet with index α = (α, . . . , α) (0 < α < 1), we prove the following uniform Hausdorff dimension result for its image sets: If N ≤ αd, then with probability one,(3) We provide sufficient conditions for the image B H (E) to be a Salem set or to have interior points. The results in (2) and (3)

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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 $B^H$.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP131 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

Body surface Laplacian electrocardiogram (ECG) mapping using a set of disk electrodes is explored by both computer simulation and human experiments in 12 healthy subjects. The Laplacian ECG was estimated from body surface potentials using finite difference estimation algorithms. The performance of the finite difference Laplacian estimators was evaluated by both computer simulation and human experiments. The present experimental results show that the two types of finite difference Laplacian estimates are highly correlated and have a consistent spatial distribution over the anterolateral chest during normal ventricular activation. The present computer simulation and human experiment results suggest the feasibility of estimating the body surface Laplacian maps (BSLMs) from potentials using the finite difference algorithm over the anterior chest in male subjects. The noise levels of the BSLMs over the anterolateral chest were quantitatively compared to the noise levels in corresponding body surface potential maps (BSPMs) in 12 healthy subjects. The simulation and experiment results indicate that the noise to signal ratios in the BSLMs over the anterolateral chest during ventricular activation is about 5 times that of the BSPMs, when no signal processing is performed.

Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\in{\mathbb{R}}_+^N$ with values in $(0,1)^N$. Let $\{B^{H(t)}(t)\}=\{B^{H(t)}(t),t\in{\mathbb{R}}^N_+\}$ be an $(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$. Under some regularity conditions on the function $H(t)$, we prove the existence, joint continuity and the H\"{o}lder regularity of the local times of $\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the level sets of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ126 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm