Comptes Rendus
no. 4
Partial differential equations/Probability theory
A Schauder estimate for stochastic PDEs
[Une estimée de Schauder pour des EDPs stochastiques]

Présenté par : the Editorial Board

Kai Du1 ; Jiakun Liu1
1 Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 371-375.

Nous considérons des équations aux dérivées partielles stochastiques, du type parabolique et à coefficients aléatoires dans des espaces de Hölder à valeurs vectorielles. Nous obtenons une estimée de Schauder optimale, puis nous utilisons cette estimée pour prouver l'existence et l'unicité de la solution du problème de Cauchy.

Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued Hölder spaces, we establish a sharp Schauder theory. The existence and uniqueness of solutions to the Cauchy problem is obtained.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.01.010
Kai Du 1 ; Jiakun Liu 1

1 Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia 
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Kai Du; Jiakun Liu. A Schauder estimate for stochastic PDEs. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 371-375. doi : 10.1016/j.crma.2016.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.010/

[1] G. Da Prato; J. Zabczyk Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992

[2] R. Dalang; D. Khoshnevisan; E. Nualart Hitting probabilities for systems of non-linear stochastic heat equations with additive noise, ALEA Lat. Am. J. Probab. Math. Stat., Volume 3 (2007), pp. 231-271

[3] K. Du; J. Liu On the Cauchy problem for stochastic parabolic equations in Hölder spaces (submitted for publication, available at) | arXiv

[4] D. Gilbarg; N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer, 2001

[5] N.V. Krylov A W2n-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Relat. Fields, Volume 98 (1994) no. 3, pp. 389-421

[6] N.V. Krylov Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996

[7] N.V. Krylov On Lp-theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal., Volume 27 (1996) no. 2, pp. 313-340

[8] N.V. Krylov An analytic approach to SPDEs, Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, vol. 64, 1999, pp. 185-242

[9] N.V. Krylov; B. Rozovsky On the Cauchy problem for linear stochastic partial differential equations, Izv. Math., Volume 11 (1977) no. 6, pp. 1267-1284

[10] R. Mikulevicius On the Cauchy problem for parabolic SPDEs in Holder classes, Ann. Probab. (2000), pp. 74-103

[11] E. Pardoux Équations aux dérivées partielles stochastiques non linéaires monotones, Univ. Paris-Sud, Orsay, 1975 (Thèse)

[12] B. Rozovsky On stochastic partial differential equations, Sb. Math., Volume 25 (1975) no. 2, pp. 295-322

[13] B. Rozovsky Stochastic Evolution Systems, Springer, 1990

[14] X.-J. Wang Schauder estimates for elliptic and parabolic equations, Chin. Ann. Math., Ser. B, Volume 27 (2006) no. 6, pp. 637-642

[15] M. Zakai On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 11 (1969) no. 3, pp. 230-243

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