A Schauder estimate for stochastic PDEs

Kai Du; Jiakun Liu

doi:10.1016/j.crma.2016.01.010

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 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

Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 371-375.

Résumé
Abstract

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 : 2015-09-18
Accepté le : 2016-01-25
Publié le : 2016-02-11

DOI : 10.1016/j.crma.2016.01.010

Affiliations des auteurs :

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

@article{CRMATH_2016__354_4_371_0,
     author = {Kai Du and Jiakun Liu},
     title = {A {Schauder} estimate for stochastic {PDEs}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {371--375},
     publisher = {Elsevier},
     volume = {354},
     number = {4},
     year = {2016},
     doi = {10.1016/j.crma.2016.01.010},
     language = {en},
}

TY  - JOUR
AU  - Kai Du
AU  - Jiakun Liu
TI  - A Schauder estimate for stochastic PDEs
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 371
EP  - 375
VL  - 354
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2016.01.010
LA  - en
ID  - CRMATH_2016__354_4_371_0
ER  -

%0 Journal Article
%A Kai Du
%A Jiakun Liu
%T A Schauder estimate for stochastic PDEs
%J Comptes Rendus. Mathématique
%D 2016
%P 371-375
%V 354
%N 4
%I Elsevier
%R 10.1016/j.crma.2016.01.010
%G en
%F CRMATH_2016__354_4_371_0

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/

Bibliographie
Cité par

[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 $W_{2}^{n}$ -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 $L_{p}$ -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

Cité par Sources :

Commentaires - Politique