On the monofractality of many stationary continuous Gaussian fields
Résumé
In this article we focus on a general real-valued continuous stationary Gaussian field X characterized by its spectral density |g| 2 , where g is any even realvalued deterministic square integrable function. Our starting point consists in drawing a close connection between critical Besov regularity of the inverse Fourier transform of g and α X the random pointwise Hölder exponent function of X, which measures local roughness/smoothness of its sample paths at each point. Then, thanks to Littlewood-Paley methods and Hausdorff-Young inequalities, under weak conditions on g, we show that the random function α X is actually a deterministic constant which does not change from point to point. This result means that the field X is of monofractal nature. Also, it is worth mentioning that such a result can easily be extended to the case where X is no longer stationary but has stationary increments.
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