English
Asymptotics for some non-linear stochastic heat equations
Multi angle
Auteurs : Nualart, Eulalia (Auteur de la Conférence)
CIRM (Editeur )
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
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 : \[
\frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in \mathbb{R}^d.
\]
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).
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
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 22/05/2018 Date de captation : 16/05/2018 Sous collection : Research talks arXiv category : Probability ; Analysis of PDEs Domaine : Probability & Statistics ; PDE Format : MP4 (.mp4) - HD Durée : 00:39:30 Audience : Researchers Download : https://videos.cirm-math.fr/2018-05-16_Nualart.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Stochastic partial differential equations / Equations aux dérivées partielles stochastiquesOrganisateurs 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
Données de citation
DOI : 10.24350/CIRM.V.19402003Citer cette vidéo: 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
- [Multi angle] Stochastic solutions of 2D fluids / Auteur de la Conférence Flandoli, Franco.
- [ Post-edited] Singular SPDE with rough coefficients / Auteur de la Conférence Otto, Felix.
- [Multi angle] Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space / Auteur de la Conférence Hofmanova, Martina.
- [Multi angle] Bessel-like SPDEs / Auteur de la Conférence Zambotti, Lorenzo.
Bibliographie
- 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
- Chen, L., & Huang, J. (2016). Comparison principle for stochastic heat equation on $\mathbb {R}^ d$.
- 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
- 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
- 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
- Foondun, M., Li, S.-T., & Joseph, M. (2016). An approximation result for a class of stochastic heat equations with colored noise.
- 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
