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Recent Advances Related to SPDEs with Fractional Noise

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Book cover Seminar on Stochastic Analysis, Random Fields and Applications VII

Part of the book series: Progress in Probability ((PRPR,volume 67))

Abstract

We review the literature related to stochastic partial differential equations with spatially-homogeneous Gaussian noise, and explain how one can introduce the structure of the fractional Brownian motion into the temporal component of the noise. The Hurst parameter H is assumed to be greater than 1/2. In the case of linear equations, we revisit the conditions for the existence of a mild solution. In the nonlinear case, we point out what are the difficulties due to the fractional component of the noise. These difficulties can be avoided in the case of equations with multiplicative noise, since in this case, the solution has a known Wiener chaos decomposition. Finally, this methodology is applied to the wave equation (in arbitrary dimension d ≥ 1), driven by a Gaussian noise which has a spatial covariance structure given by the Riesz kernel.

Mathematics Subject Classification (2010). Primary 60H15; Secondary 60H07.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario, K1N 6N5, Canada

    Raluca M. Balan

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  1. Raluca M. Balan
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Correspondence to Raluca M. Balan .

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Editors and Affiliations

  1. Institut de Mathématiques (IMA) Chaire de Statistiques Appliquées (STAP), Ecole Polytechnique Fédérale de Lausanne Fac. Sciences de Base (FSB), Lausanne, Switzerland

    Robert C. Dalang

  2. Institut Elie Cartan, Université de Lorraine, Vandoeuvre-les-Nancy CX, France

    Marco Dozzi

  3. Unité de Mathématiques appliquées, ENSTA ParisTech, Palaiseau, France

    Francesco Russo

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Balan, R.M. (2013). Recent Advances Related to SPDEs with Fractional Noise. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VII. Progress in Probability, vol 67. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0545-2_1

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