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Degenerate parabolic stochastic partial differential equations

Martina Hofmanová 1, 2 
2 IPSO - Invariant Preserving SOlvers
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyperbolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochastic forcing and study their approximations in the sense of Bhatnagar-Gross-Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkhod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition.
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Submitted on : Tuesday, December 10, 2013 - 2:30:34 PM
Last modification on : Friday, May 20, 2022 - 9:04:52 AM
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  • HAL Id : tel-00916580, version 1


Martina Hofmanová. Degenerate parabolic stochastic partial differential equations. General Mathematics [math.GM]. École normale supérieure de Cachan - ENS Cachan; Charles University. Faculty of mathematics and physics. Department of metal physics (Prague), 2013. English. ⟨NNT : 2013DENS0024⟩. ⟨tel-00916580⟩



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