Abstract
We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.
Nous utilisons le calcul paracontrôlé pour étudier le comportement asymptotique de certaines EDP quasi-linéaires avec bruit régularisé, qui sont apparues initialement comme limites d’échelle spatio-temporelles de systèmes de particules en environnement aléatoire sur le réseau discret un-dimensionnel. Nous établissons un résultat de convergence et montrons que l’EDP stochastique limite est bien posée localement en temps avec un bruit blanc en espace. Il apparaît que notre EDP stochastique limite ne nécessite pas de renormalisation. Nous donnons aussi un théorème de comparaison pour l’équation limite.
Acknowledgements
T. Funaki was supported in part by JSPS KAKENHI, Grant-in-Aid for Scientific Researches (A) 18H03672 and (S) 16H06338. M. Hoshino was supported in part by JSPS KAKENHI, Early-Career Scientists 19K14556. S. Sethuraman was supported by grant ARO W911NF-181-0311, a Simons Foundation Sabbatical grant, and by a Japan Society for the Promotion of Science Fellowship. B. Xie was supported in part by JSPS KAKENHI, Grant-in-Aid for Scientific Research (C) 16K05197 and (C) 20K03627.
Citation
Tadahisa Funaki. Masato Hoshino. Sunder Sethuraman. Bin Xie. "Asymptotics of PDE in random environment by paracontrolled calculus." Ann. Inst. H. Poincaré Probab. Statist. 57 (3) 1702 - 1735, August 2021. https://doi.org/10.1214/20-AIHP1129
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