A simple method for finite range decomposition of quadratic forms and Gaussian fields

TL;DR

This paper introduces a straightforward technique to decompose Green forms associated with symmetric Dirichlet forms into finite-range, smoother components, enabling multiscale analysis of Gaussian free fields with localized correlations.

Contribution

The authors develop a simple, wave-equation-based method for finite range decomposition of quadratic forms, improving existing results and simplifying proofs.

Findings

Enables multiscale decomposition of Gaussian free fields

Provides finite range, smoother Green form components

Utilizes wave equation propagation and Chebyshev polynomials

Abstract

We present a simple method to decompose the Green forms corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green form. This result gives rise to multiscale decompositions of the associated Gaussian free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Our method makes use of the finite propagation speed of the wave equation and Chebyshev polynomials. It improves several existing results and also gives simpler proofs.