Local times of fractional Brownian sheets
Consider a real-valued fractional Brownian sheet \(B^{H}_{0}=\{ B^{H}_{0}(t)\), \(t\in \mathbb{R}^{N}_{+} \}\). Let \(B^{H}\) be the \((N,d)\) Gaussian random field defined by \(B^{H}(t)=(B^{H}_{1}(t), \ldots , B^{H}_{d}(t))\) \((t\in \mathbb{R}^{N}_{+})\), where \(B^{H}_{1}, \ldots , B^{H}_{d}\) are independent copies of \(B^{H}_{0}.\) The authors show the existence and joint continuity of the local times of the Gaussian random field \(B^{H}.\) Many authors have studied the sample path properties of the Brownian sheet and fractional Brownian motion. And most of the studies are established by the facts of the independent increment property of the Brownian sheet and the local nondeterminism of fractional Brownian motion. But, in general, the fractional Brownian sheet \({B}^{H}\) has neither the property of independent increments nor local nondeterminism. Some researchers have investigated the properties of the fractional Brownian motion without independent increment and local nondeterminism. The authors claim that the existence theorem is sharp and its proof is quite different from the proofs for the Brownian sheet and fractional Brownian motion. However, for the joint continuity, the authors only establish a sufficient condition.
scientific article; zbMATH DE number 1837375