Finite range decomposition for a general class of elliptic operators

TL;DR

This paper establishes a uniform finite range decomposition for covariance operators of gradient Gaussian vector fields on ^d with non-translation-invariant covariance, providing optimal regularity bounds and extending previous results.

Contribution

It introduces a finite range decomposition for a broad class of elliptic operators with non-translation-invariant covariance, along with regularity bounds, extending prior work.

Findings

Finite range decomposition for non-translation-invariant covariance operators.

Optimal regularity bounds for subcovariance operators.

Extension of previous results to more general elliptic operators.

Abstract

We consider a family of gradient Gaussian vector fields on $\Z^d$, where the covariance operator is not translation invariant. A uniform finite range decomposition of the corresponding covariance operators is proven, i.e., the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of increasing diameter. An optimal regularity bound for the subcovariance operators is proven. We also obtain regularity bounds as we vary the coefficients defining the gradient Gaussian measures. This extends a result of S. Adams, R. Koteck\'y and S. M\"uller \cite{1202.1158}.