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May 2020 Stochastic Hölder continuity of random fields governed by a system of stochastic PDEs
Kai Du, Jiakun Liu, Fu Zhang
Ann. Inst. H. Poincaré Probab. Statist. 56(2): 1230-1250 (May 2020). DOI: 10.1214/19-AIHP1000


This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in $L^{p}$-space of random variables. A modified stochastic parabolicity condition involving $p$ is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.

Cet article construit une théorie sur la solvabilité d’un système d’équations différentielles partielles stochastiques. En raison du théorème de continuité de Kolmogorov, les solutions sont recherchées dans certaines classes de Hölder, dans lesquelles un champ aléatoire est considéré comme une fonction spatio-temporelle prenant des valeurs dans l’espace $L^{p}$ des variables aléatoires. Une condition de parabolicité stochastique modifiée impliquant $p$ est proposée afin d’assurer la finitude de la norme associée de la solution. En étudiant des exemples, cette condition est montrée être optimale. Les estimations de type de Schauder et la solvabilité de l’équation sont démontrées.


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Kai Du. Jiakun Liu. Fu Zhang. "Stochastic Hölder continuity of random fields governed by a system of stochastic PDEs." Ann. Inst. H. Poincaré Probab. Statist. 56 (2) 1230 - 1250, May 2020.


Received: 19 October 2018; Revised: 10 April 2019; Accepted: 2 May 2019; Published: May 2020
First available in Project Euclid: 16 March 2020

zbMATH: 07199896
MathSciNet: MR4076782
Digital Object Identifier: 10.1214/19-AIHP1000

Primary: 35R60 , 60H15
Secondary: 35K45

Keywords: Schauder estimate , Stochastic continuity , Stochastic parabolicity condition , Stochastic partial differential system

Rights: Copyright © 2020 Institut Henri Poincaré


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Vol.56 • No. 2 • May 2020
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