Revista Matemática Iberoamericana


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Volume 13, Issue 1, 1997, pp. 19–90
DOI: 10.4171/RMI/217

Elliptic gaussian random processes

Albert Benassi[1], Daniel Roux[2] and Stéphane Jaffard[3]

(1) Laboratoire de Mathématiques, CNRS UPRESA 6016, Université Blaise Pascal, Clermont-Ferrand 2, 63177, AUBIÈRE CEDEX, FRANCE
(2) Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, Clermont-Ferrand 2, 63177, AUBIÈRE CEDEX, FRANCE
(3) Laboratoire d'Analyse et de Mathématiques Appliqué, Université Paris Est, 61, avenue du Général de Gaulle, 94010, CRÉTEIL CEDEX, FRANCE

We study the Gaussian random fields indexed by $\mathbb R^d$ whose covariance is defined in all generality as the parametrix of an elliptic pseudodifferential operator with minimal regularity assumption on the symbol. We construct new wavelet bases adapted to these operators; the decomposition of the field on this corresponding basis yields its iterated logarithm law and its uniform modulus of continuity. We also characterize the local scalings of the field in term of the properties of the principal symbol of the pseudodifferential operator. Similar results are obtained for the Multi-Fractional Brownian Motion.

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Benassi A, Roux D, Jaffard S. Elliptic gaussian random processes. Rev. Mat. Iberoamericana 13 (1997), 19-90. doi: 10.4171/RMI/217