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H 1 Asymptotics for some non-linear stochastic heat equations

Auteurs : Nualart, Eulalia (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Consider the following stochastic heat equation,
    \[
    \frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in \mathbb{R}^d.
    \]
    Here $-\nu(-\Delta)^{\alpha/2}$ is the fractional Laplacian with $\nu>0$ and $\alpha \in (0,2]$, $\sigma: \mathbb{R}\rightarrow \mathbb{R}$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable conditions, we will explore the effect of the initial data on the spatial asymptotic properties of the solution. We also prove a strong comparison principle thus filling an important gap in the literature.
    Joint work with Mohammud Foondun (University of Strathclyde).

    Codes MSC :
    35R60 - PDEs with randomness, stochastic PDE
    60H15 - Stochastic partial differential equations
    60J55 - Local time and additive functionals

    Ressources complémentaires :
    https://www.cirm-math.fr/ProgWeebly/Renc1742/Nualart.pdf

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 22/05/2018
      Date de captation : 16/05/2018
      Collection : Research talks ; Partial Differential Equations ; Probability and Statistics
      Format : MP4
      Durée : 00:39:30
      Domaine : Probability & Statistics ; PDE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2018-05-16_Nualart.mp4

    Informations sur la rencontre

    Nom de la rencontre : Stochastic partial differential equations / Equations aux dérivées partielles stochastiques
    Organisateurs de la rencontre : Berglund, Nils ; Debussche, Arnaud ; Delarue, François ; Kuehn, Christian
    Dates : 14/05/2018 - 18/05/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1742.html

    Citation Data

    DOI : 10.24350/CIRM.V.19402003
    Cite this video as: Nualart, Eulalia (2018). Asymptotics for some non-linear stochastic heat equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19402003
    URI : http://dx.doi.org/10.24350/CIRM.V.19402003


    Voir aussi

    Bibliographie

    1. Chen, L., Khoshnevisan, D., & Kim, K. (2017). A boundedness trichotomy for the stochastic heat equation. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques, 53(4), 1991-2004 - https://doi.org/10.1214/16-AIHP780

    2. Chen, L., & Huang, J. (2016). Comparison principle for stochastic heat equation on $\mathbb {R}^ d$. - https://arxiv.org/abs/1607.03998

    3. Chen, L., & Kim, K. (2017). On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques, 53(1), 358-388 - http://dx.doi.org/10.1214/15-AIHP719

    4. Chen, L. (2016). Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise. The Annals of Probability, 44(2), 1535-1598 - http://dx.doi.org/10.1214/15-AOP1006

    5. Conus, D., Joseph, M., & Khoshnevisan, D. (2013). On the chaotic character of the stochastic heat equation, before the onset of intermitttency. The Annals of Probability, 41(3B), 2225-2260 - https://doi.org/10.1214/11-AOP717

    6. Foondun, M., Li, S.-T., & Joseph, M. (2016). An approximation result for a class of stochastic heat equations with colored noise. - https://arxiv.org/abs/1611.06829

    7. Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stochastics and Stochastic Reports, 37(4), 225-245 - https://doi.org/10.1080/17442509108833738

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