Local times of multifractional Brownian sheets

Abstract

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 Hölder 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.

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)

References in corpus (3)

• Joint continuity of the local times of fractional Brownian sheets
• Sample path properties of the local time of multifractional Brownian motion
• Fractional Brownian fields, duality, and martingales

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• Approximation to multifractional Riemann-Liouville Brownian sheet
• Local times for multifractional Brownian motion in higher dimensions: A white noise approach
• Some sample path properties of multifractional Brownian motion
• Uniformly and strongly consistent estimation for the Hurst function of a Linear Multifractional Stable Motion
• Self-intersection local times for generalized grey Brownian motion in higher dimensions