Local times of multifractional Brownian sheets

Abstract

Denote by $H(t)=(H_1(t), …, H_N(t))$ a function in $t∈ℝ_+^N$ with values in $(0, 1)^N$. Let $\{B^{H(t)}(t)\}=\{B^{H(t)}(t), t∈ℝ_+^N\}$ 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.