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Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method

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2021

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Dhariwal, Gaurav
Huber, Florian
Jüngel, Ansgar
Kuehn, Christian

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Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. Gauthier-Villars. 2021, 57(1), pp. 577-602. ISSN 0246-0203. eISSN 1778-7017. Available under: doi: 10.1214/20-AIHP1088

Zusammenfassung

The existence of global-in-time bounded martingale solutions to a general class of cross-diffusion systems with multiplicative Stratonovich noise is proved. The equations describe multicomponent systems from physics or biology with volume-filling effects and possess a formal gradient-flow or entropy structure. This structure allows for the derivation of almost surely positive lower and upper bounds for the stochastic processes. The existence result holds under some assumptions on the interplay between the entropy density and the multiplicative noise terms. The proof is based on a stochastic Galerkin method, a Wong–Zakai type approximation of the Wiener process, the boundedness-by-entropy method, and the tightness criterion of Brzeźniak and coworkers. Three-species Maxwell–Stefan systems and n-species biofilm models are examples that satisfy the general assumptions.

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510 Mathematik

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Biofilm model , Cross diffusion , entropy method , Martingale solutions , Maxwell–Stefan systems , Skorokhod–Jakubowski theorem , tightness

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ISO 690DHARIWAL, Gaurav, Florian HUBER, Ansgar JÜNGEL, Christian KUEHN, Alexandra BLESSING-NEAMTU, 2021. Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method. In: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. Gauthier-Villars. 2021, 57(1), pp. 577-602. ISSN 0246-0203. eISSN 1778-7017. Available under: doi: 10.1214/20-AIHP1088
BibTex
@article{Dhariwal2021Globa-57998,
  year={2021},
  doi={10.1214/20-AIHP1088},
  title={Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method},
  number={1},
  volume={57},
  issn={0246-0203},
  journal={Annales de l'Institut Henri Poincaré, Probabilités et Statistiques},
  pages={577--602},
  author={Dhariwal, Gaurav and Huber, Florian and Jüngel, Ansgar and Kuehn, Christian and Blessing-Neamtu, Alexandra}
}
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