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

Abstract The statistical properties of non-linear observables of the fractional Gaussian field ϕ ( 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 ϕ n ( x → ) of the ill-defined composite operators ϕ n ( x → ) . For the special case n = 2 of quadratic observables, explicit results include the cumulants of arbitrary order, the Lévy–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 ϕ n ( x → ) of the ill-defined composite operators ϕ n ( x → ) and to compute their correlations involving the Hurst exponents H n = n H .