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Stochastic Analysis: A Series of LecturesStochastic Porous Media Equations

Stochastic Analysis: A Series of Lectures: Stochastic Porous Media Equations [This survey is devoted to the presentation of a few recent results concerning the existence, longtime behaviour and localization of solutions to stochastic porous media equations with linearly multiplicative Gaussian noise. Some of these results are given without proof, or with a sketch of proof only. In Section 1, we present a few basic results on existence of solutions and some models described by deterministic porous media equations. Section 2 is devoted to an existence and uniqueness result for porous media equations perturbed by a linear multiplicative Wiener process. In Section 3, we present two results pertaining to extinction in finite time of solutions to fast diffusion porous media equations with linear multiplicative Gaussian noise and to propagation with finite speed in the low diffusion case. Section 4 is devoted to some asymptotic results for the solutions of stochastic self-organized criticality equations.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Stochastic Analysis: A Series of LecturesStochastic Porous Media Equations

Part of the Progress in Probability Book Series (volume 68)
Editors: Dalang, Robert C.; Dozzi, Marco; Flandoli, Franco; Russo, Francesco
Barbu, Viorel
Stochastic Analysis: A Series of Lectures — May 15, 2015
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Publisher
Springer Basel
Copyright
© Springer Basel 2015
ISBN
978-3-0348-0908-5
Pages
101 –133
DOI
10.1007/978-3-0348-0909-2_4
Publisher site
See Chapter on Publisher Site

Abstract

[This survey is devoted to the presentation of a few recent results concerning the existence, longtime behaviour and localization of solutions to stochastic porous media equations with linearly multiplicative Gaussian noise. Some of these results are given without proof, or with a sketch of proof only. In Section 1, we present a few basic results on existence of solutions and some models described by deterministic porous media equations. Section 2 is devoted to an existence and uniqueness result for porous media equations perturbed by a linear multiplicative Wiener process. In Section 3, we present two results pertaining to extinction in finite time of solutions to fast diffusion porous media equations with linear multiplicative Gaussian noise and to propagation with finite speed in the low diffusion case. Section 4 is devoted to some asymptotic results for the solutions of stochastic self-organized criticality equations.]

Published: May 15, 2015

Keywords: Wiener process; porous media equation; finite time extinction; selforganized criticality; Primary 60H15; Secondary 35R60; 76M35

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