Tangent fields, intrinsic stationarity, and self-similarity

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 the seminal work of Falconer (2002), we show that the generalized $k$-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron (1973). 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 (2009) and Berschneider (2012), we establish the spectral characterization of all second order ${\mathbb V}$-valued IRF$_k$'s, extending the classic Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar ${\mathbb V}$-valued IRF$_k$'s, extending some results of Dobrushin (1979) and Didier, Meerschaert, and Pipiras (2018), among others. These processes are ${\mathbb V}$-valued counterparts to $k$-order 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. In a technical Supplement of independent interest, we provide a unified treatment of the spectral theory for second-order stationary and intrinsically stationary processes taking values in a separable Hilbert space. We give the proofs of the Bochner-Neeb and Bochner-Schwartz theorems.