IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v119y2016icp310-316.html
   My bibliography  Save this article

SPDEs with rough noise in space: Hölder continuity of the solution

Author

Listed:
  • Balan, Raluca M.
  • Jolis, Maria
  • Quer-Sardanyons, Lluís

Abstract

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than H, and Hölder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:119:y:2016:i:c:p:310-316
    DOI: 10.1016/j.spl.2016.09.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715216301687
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2016.09.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Giordano, Luca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2020. "SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7396-7430.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    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. 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.
    4. Rama Cont, 1999. "Modeling interest rate dynamics: an infinite-dimensional approach," Papers cond-mat/9902018, arXiv.org.
    5. 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.
    6. 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.
    7. Lund, Adam & Hansen, Niels Richard, 2019. "Sparse network estimation for dynamical spatio-temporal array models," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    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. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:119:y:2016:i:c:p:310-316. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.