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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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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