Stochastic partial differential equations / Equations aux dérivées partielles stochastiques

Collection Stochastic partial differential equations / Equations aux dérivées partielles stochastiques

Organizer(s) Berglund, Nils ; Debussche, Arnaud ; Delarue, François ; Kuehn, Christian

Date(s) 14/05/2018 - 18/05/2018

linked URL https://conferences.cirm-math.fr/1742.html

00:00:00 / 00:00:00

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Asymptotics for some non-linear stochastic heat equations

By Eulalia Nualart

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).

Information about the video

Date of recording 16/05/2018
Date of publication 22/05/2018
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19402003
Cite this video Nualart, Eulalia (16/05/2018). Asymptotics for some non-linear stochastic heat equations. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19402003
URL https://dx.doi.org/10.24350/CIRM.V.19402003

Domain(s)

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 $\mathbb {R}^ d$. <arXiv:1607.03998> - https://arxiv.org/abs/1607.03998
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. <arXiv:1611.06829> - https://arxiv.org/abs/1611.06829
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

MSC codes

Document(s)

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

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