Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation |
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Authors: | S. Tindel |
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Affiliation: | a Département de Mathématiques, Institut Galilée, Université de Paris 13, Avenue J.-B. Clément, 93430-Villetaneuse, France b Laboratoire de Probabilités, Université de Paris 6, 4, Place Jussieu, 75252 Paris Cedex 05, France c Department of Statistics & Department of Mathematics, Purdue University, 1399 Math Sci. Bldg., 150 N. University St. West Lafayette, IN 47907-2067, USA |
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Abstract: | A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given. |
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Keywords: | primary 60G15 secondary 60G17 60H15 42A16 |
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