Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators

TL;DR

This paper investigates the statistical behavior of non-linear and composite observables of fractal Gaussian fields with negative Hurst exponents, providing explicit formulas and analysis for spatial-averaging and higher-order operators.

Contribution

It introduces a detailed analysis of the finite parts of ill-defined composite operators in fractal Gaussian fields, including explicit results for quadratic observables and a Wiener-Ito chaos expansion for higher orders.

Findings

Explicit cumulants and characteristic functions for quadratic observables

Analysis of large deviations properties of the field

Identification of finite parts of composite operators using stochastic integrals

Abstract

The statistical properties of non-linear observables of the fractal Gaussian field $\phi(\vec x)$ of negative Hurst exponent $H<0$ in dimension $d$ are revisited with a focus on spatial-averaging observables and on the properties of the finite parts $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the L\'evy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order $n>2$ is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$.