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March 2014 Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise
Benjamin Gess
Ann. Probab. 42(2): 818-864 (March 2014). DOI: 10.1214/13-AOP869

Abstract

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in $L^{1}(\mathcal{O})$ on bounded domains $\mathcal{O} $. The generation of a continuous, order-preserving random dynamical system on $L^{1}(\mathcal{O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^{\infty}(\mathcal{O})$ norm. Uniform $L^{\infty}$ bounds and uniform space–time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated and a pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space–time rough signals, existence of solutions is proven for initial data in $L^{m+1}(\mathcal{O})$.

Citation

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Benjamin Gess. "Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise." Ann. Probab. 42 (2) 818 - 864, March 2014. https://doi.org/10.1214/13-AOP869

Information

Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 06288295
MathSciNet: MR3178475
Digital Object Identifier: 10.1214/13-AOP869

Subjects:
Primary: 37L55 , 76S05
Secondary: 37L30 , 60H15

Keywords: random attractor , Random dynamical system , stochastic fast diffusion equation , Stochastic partial differential equations , stochastic porous medium equation , Wong–Zakai approximation

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 2 • March 2014
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