Tangent fields, intrinsic stationarity, and self similarity

Abstract

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${}_k$, $k=0,1,\dots$). 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-valued versions of the general k-th order operator fractional Brownian fields and are characterized by their self-similarity operator exponent as well as a finite trace class operator valued spectral measure. We conclude with several examples motivating future applications to probability and statistics.