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.
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.
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Kai Du 1; Jiakun Liu 1
@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}, }
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] Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992
[2] 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] On the Cauchy problem for stochastic parabolic equations in Hölder spaces (submitted for publication, available at) | arXiv
[4] Elliptic Partial Differential Equations of Second Order, Springer, 2001
[5] A -theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Relat. Fields, Volume 98 (1994) no. 3, pp. 389-421
[6] Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996
[7] On -theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal., Volume 27 (1996) no. 2, pp. 313-340
[8] An analytic approach to SPDEs, Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, vol. 64, 1999, pp. 185-242
[9] On the Cauchy problem for linear stochastic partial differential equations, Izv. Math., Volume 11 (1977) no. 6, pp. 1267-1284
[10] On the Cauchy problem for parabolic SPDEs in Holder classes, Ann. Probab. (2000), pp. 74-103
[11] Équations aux dérivées partielles stochastiques non linéaires monotones, Univ. Paris-Sud, Orsay, 1975 (Thèse)
[12] On stochastic partial differential equations, Sb. Math., Volume 25 (1975) no. 2, pp. 295-322
[13] Stochastic Evolution Systems, Springer, 1990
[14] Schauder estimates for elliptic and parabolic equations, Chin. Ann. Math., Ser. B, Volume 27 (2006) no. 6, pp. 637-642
[15] On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 11 (1969) no. 3, pp. 230-243
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