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Asymptotics for some non-linear stochastic heat equations
Multi angle
Authors : Nualart, Eulalia (Author of the conference)
CIRM (Publisher )
35R60 - PDEs with randomness, stochastic PDE
60H15 - Stochastic partial differential equations
60J55 - Local time and additive functionals
Additional resources :
https://www.cirm-math.fr/ProgWeebly/Renc1742/Nualart.pdf
CIRM (Publisher )
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Abstract :
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).
MSC Codes : \[
\frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in \mathbb{R}^d.
\]
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
Additional resources :
https://www.cirm-math.fr/ProgWeebly/Renc1742/Nualart.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 22/05/2018 Conference Date : 16/05/2018 Subseries : Research talks arXiv category : Probability ; Analysis of PDEs Mathematical Area(s) : Probability & Statistics ; PDE Format : MP4 (.mp4) - HD Video Time : 00:39:30 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2018-05-16_Nualart.mp4 
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Information on the Event
Event Title : Stochastic partial differential equations / Equations aux dérivées partielles stochastiquesEvent Organizers : Berglund, Nils ; Debussche, Arnaud ; Delarue, François ; Kuehn, Christian Dates : 14/05/2018 - 18/05/2018 Event Year : 2018 Event URL : https://conferences.cirm-math.fr/1742.html
Citation Data
DOI : 10.24350/CIRM.V.19402003Cite 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 |
See Also
- [Multi angle] Stochastic solutions of 2D fluids / Author of the conference Flandoli, Franco.
- [Post-edited] Singular SPDE with rough coefficients / Author of the conference Otto, Felix.
- [Multi angle] Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space / Author of the conference Hofmanova, Martina.
- [Multi angle] Bessel-like SPDEs / Author of the conference Zambotti, Lorenzo.
Bibliography
- 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
