Tangent fields, intrinsic stationarity, and self-similarity (with a supplement on Matheron Theory)

This paper studies the local structure of continuous random fields on $mathbb R^d$ taking values in a complete separable linear metric space ${mathbb V}$. Extending seminal work of Falconer, we show that the generalized $(1+k)$-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of ${mathbb V}$-valued intrinsic random functions of order $k$ (IRF$_k$, $k=0,1,cdots$). To this end, we focus on the special case where ${mathbb V}$ is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order ${mathbb V}$-valued IRF$_k$s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar ${mathbb V}$-valued IRF$_k$s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valu