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On stochastic partial differential equations with spatially correlated noise: smoothness of the law

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  • Márquez-Carreras, D.
  • Mellouk, M.
  • Sarrà, M.

Abstract

We deal with the following general kind of stochastic partial differential equations:with null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u(t,x) possesses a smooth density pt,x for every . We use the tools of Malliavin Calculus. Secondly, we apply this general result to two particular cases: the d-dimensional spatial heat equation, d[greater-or-equal, slanted]1, and the wave equation, d[set membership, variant]{1,2}.

Suggested Citation

  • Márquez-Carreras, D. & Mellouk, M. & Sarrà, M., 2001. "On stochastic partial differential equations with spatially correlated noise: smoothness of the law," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 269-284, June.
    • Handle: RePEc:eee:spapps:v:93:y:2001:i:2:p:269-284
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    References listed on IDEAS

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    1. Peszat, Szymon & Zabczyk, Jerzy, 1997. "Stochastic evolution equations with a spatially homogeneous Wiener process," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 187-204, December.
    2. Millet, Annie & Morien, Pierre-Luc, 2000. "On a stochastic wave equation in two space dimensions: regularity of the solution and its density," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 141-162, March.
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    Cited by:

    1. Ferrario, Benedetta & Zanella, Margherita, 2019. "Absolute continuity of the law for the two dimensional stochastic Navier–Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1568-1604.
    2. Nualart, David & Quer-Sardanyons, Lluís, 2009. "Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3914-3938, November.

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