A characterisation of the continuum Gaussian free field in $d \geq 2$ dimensions

TL;DR

This paper characterizes the $d$-dimensional Gaussian free field as the unique process with translation invariance, scaling, and domain Markov property, using harmonic analysis techniques.

Contribution

It provides a uniqueness characterization of the Gaussian free field in higher dimensions based on invariance and Markov properties.

Findings

Proves the Gaussian free field is unique under specified conditions.

Uses harmonic analysis to decompose the functional space.

Establishes foundational properties of the Gaussian free field in $d geq 2$.

Abstract

We prove that under certain mild moment and continuity assumptions, the $d$-dimensional Gaussian free field is the only stochastic process in $d\geq 2$ that is translation invariant, exhibits a certain scaling, and satisfies the usual domain Markov property. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.