On local times of self-similar random fields
The author considers \(\kappa\)-self-similar stochastic fields \(X: \Omega\times \mathbb{R}^N\to \mathbb{R}^d\), i.e. such fields that \(X(c\cdot)\) and \(c^\kappa X(\cdot)\) have the same distribution. In addition, for any \(m\geq 1\) the distribution of the random vector \(\{X(t_1+ s_1)- X(t_1),\dots, X(t_m+ s_m)- X(t_m)\}\), \(t_j,s_j\in \mathbb{R}^N\), does not depend on \(\{t_1,\dots, t_m\}\). Under some restrictions on characteristic functions \(E\exp\{i \sum^k_{j= 1}(z_j, x(s_j))\}\), the author establishes for the field \(X\) the existence of occupation density \(\alpha(x, B)\) that belongs to \(L^2(\mathbb{R}^d, x)\) a.s. Namely, such occupation density, or local time, exists if and only if \(\kappa d< N\). The continuity and Hölder properties of \(\alpha(x, B)\) are also considered.
scientific article; zbMATH DE number 794513
scientific article