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Stochastic evolution equations with a spatially homogeneous Wiener process

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  • Peszat, Szymon
  • Zabczyk, Jerzy

Abstract

A semilinear parabolic equation on d with a non-additive random perturbation is studied. The noise is supposed to be a spatially homogeneous Wiener process. Conditions for the existence and uniqueness of the solution in terms of the spectral measure of the noise are given. Applications to population and geophysical models are indicated. The Freidlin-Wentzell large deviation estimates are obtained as well.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:72:y:1997:i:2:p:187-204
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    References listed on IDEAS

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    1. Bojdecki, Tomasz & Jakubowski, Jacek, 1989. "Ito stochastic integral in the dual of a nuclear space," Journal of Multivariate Analysis, Elsevier, vol. 31(1), pages 40-58, October.
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    Cited by:

    1. Lund, Adam & Hansen, Niels Richard, 2019. "Sparse network estimation for dynamical spatio-temporal array models," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    2. Basson, Arnaud, 2008. "Spatially homogeneous solutions of 3D stochastic Navier-Stokes equations and local energy inequality," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 417-451, March.
    3. Talarczyk, Anna, 2001. "Self-intersection local time of order k for Gaussian processes in," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 17-72, November.
    4. Duncan, T.E. & Maslowski, B. & Pasik-Duncan, B., 2005. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1357-1383, August.
    5. C. Cardon-Weber & A. Millet, 2004. "On Strongly Petrovskii's Parabolic SPDEs in Arbitrary Dimension and Application to the Stochastic Cahn–Hilliard Equation," Journal of Theoretical Probability, Springer, vol. 17(1), pages 1-49, January.
    6. Balan, Raluca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2016. "SPDEs with rough noise in space: Hölder continuity of the solution," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 310-316.
    7. Wang, JinRong, 2015. "Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 315-323.
    8. Tessitore, Gianmario & Zabczyk, Jerzy, 1998. "Strict positivity for stochastic heat equations," Stochastic Processes and their Applications, Elsevier, vol. 77(1), pages 83-98, September.
    9. Rama Cont, 1999. "Modeling interest rate dynamics: an infinite-dimensional approach," Papers cond-mat/9902018, arXiv.org.
    10. Xiangfeng Yang & Kai Yao, 2017. "Uncertain partial differential equation with application to heat conduction," Fuzzy Optimization and Decision Making, Springer, vol. 16(3), pages 379-403, September.
    11. 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.
    12. Fannjiang, Albert & Komorowski, Tomasz & Peszat, Szymon, 2002. "Lagrangian dynamics for a passive tracer in a class of Gaussian Markovian flows," Stochastic Processes and their Applications, Elsevier, vol. 97(2), pages 171-198, February.
    13. Chen, Le & Dalang, Robert C., 2015. "Moment bounds and asymptotics for the stochastic wave equation," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1605-1628.
    14. Bojdecki, Tomasz & Jakubowski, Jacek, 1999. "Invariant measures for generalized Langevin equations in conuclear space," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 1-24, November.

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