Local properties of a multifractional stable field
The author considers anisotropic harmonizable multifractional stable \((N,d)\)-fields of the form NEWLINE\[NEWLINEZ(t)=\Re\int_{\mathbb R^N}\prod_{k=1}^N\frac{e^{i\,t_kx_k}-1}{|x_k|^{1/\alpha-H_k(t)}}\,M(dx)\quad\text{ for }t\in\mathbb R^NNEWLINE\]NEWLINE with Hölder-continuous Hurst function \(H=(H_1,\dots,H_N)\) in \((0,1)^N\) and \(M=(m_1,\dots,m_d)\) for independent complex rotationally invariant symmetric \(\alpha\)-stable measures \(m_1,\dots,m_d\) on \(\mathbb R^N\) for some \(\alpha\in(1,2]\). Continuity of such fields is shown and the author gives upper and lower bounds for the increments of the field. Further, existence and regularity of the local time of the field is proven. In the Gaussian case \(\alpha=2\), a certain variant of the field is shown to have the property of sectorial local nondeterminism. As a consequence, this implies joint continuity of the local time and enables to compute the Hausdorff dimension of the image of the field.
scientific article; zbMATH DE number 6208806
scientific article