Decompositions of log-correlated fields with applications

TL;DR

This paper develops new decompositions for Gaussian fields, especially log-correlated ones, enabling extension of key results in Gaussian multiplicative chaos theory to broader classes of fields.

Contribution

It introduces two novel decomposition theorems for Gaussian fields, including a local decomposition for log-correlated fields, facilitating new applications in chaos theory.

Findings

Decomposition of Gaussian fields into H"older and independent components.

Extension of chaos existence and properties to general log-correlated fields.

Development of covariance inequalities for imaginary chaos.

Abstract

In this article we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a H\"older continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal -- or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a H\"older continuous function and an independent almost $\star$-scale invariant field (a special class of stationary log-correlated fields with 'cone-like' white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_X(x,y)=\log(1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon}$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star$-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $\beta$, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.