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| Auteur principal: | |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2505.16356 |
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Table des matières:
- The statistical properties of non-linear observables of the fractal Gaussian field $ϕ(\vec x)$ 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 $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec 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 $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$.