Supremum estimates for degenerate, quasilinear stochastic partial differential equations. (English. French summary) Zbl 1433.60052

Summary: We prove a priori estimates in \(L_{\infty }\) for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant \(\varepsilon\) and thus imply analogous estimates for degenerate quasilinear stochastic partial differential equations, such as the stochastic porous medium equation.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G46 Martingales and classical analysis
Full Text: DOI arXiv Euclid


[1] V. Barbu, G. Da Prato and M. Röckner. Michael Stochastic Porous Media Equations. Lecture Notes in Mathematics2163. Springer, Cham, 2016. · Zbl 1355.60004
[2] V. Barbu and M. Röckner. An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. J. Eur. Math. Soc. (JEMS)17 (7) (2015) 1789-1815. · Zbl 1327.60122
[3] L. A. Caffarelli and L. C. Evans. Continuity of the temperature in the two-phase Stefan problem. Arch. Ration. Mech. Anal.81 (3) (1983) 199-220. · Zbl 0516.35080 · doi:10.1007/BF00250800
[4] K. Dareiotis and M. Gerencsér. On the boundedness of solutions of SPDEs. Stoch. Partial Differ. Equ. Anal. Comput.3 (1) (2015) 84-102. · Zbl 1310.60092
[5] K. Dareiotis and M. Gerencsér. Local \(L_{\infty }\)-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs. J. Differential Equations262 (1) (2017) 615-632. · Zbl 1356.60099
[6] A. Debussche, S. de Moor and M. Hofmanová. A regularity result for quasilinear stochastic partial differential equations of parabolic type. SIAM J. Math. Anal.47 (2) (2015) 1590-1614. · Zbl 1327.60124 · doi:10.1137/130950549
[7] L. Denis, A. Matoussi and L. Stoica. \(L_p\) estimates for the uniform norm of solutions of quasilinear SPDE’s. Probab. Theory Related Fields133 (4) (2005) 437-463. · Zbl 1085.60043 · doi:10.1007/s00440-005-0436-5
[8] E. DiBenedetto. Continuity of weak solutions to a general porous medium equation. Indiana Univ. Math. J.32 (1) (1983) 83-118. · Zbl 0526.35042 · doi:10.1512/iumj.1983.32.32008
[9] E. DiBenedetto. Degenerate Parabolic Equations. Universitext. Springer-Verlag, New York, 1993. · Zbl 0794.35090
[10] M. Gerencsér. Boundary regularity of stochastic PDEs. Arxiv Preprint, arXiv:1705.05364, to appear in Annals of Probability. · Zbl 1447.60100
[11] M. Gerencsér, I. Gyöngy and N. Krylov. On the solvability of degenerate stochastic partial differential equations in Sobolev spaces. Stoch. Partial Differ. Equ. Anal. Comput.3 (1) (2015) 52-83. · Zbl 1310.60094
[12] B. Gess. Strong solutions for stochastic partial differential equations of gradient type. J. Funct. Anal.263 (8) (2012) 2355-2383. · Zbl 1267.60072 · doi:10.1016/j.jfa.2012.07.001
[13] B. Gess. Random attractors for stochastic porous media equations perturbed by space-time linear multiplicative noise. Ann. Probab.42 (2) (2014) 818-864. · Zbl 1385.37082 · doi:10.1214/13-AOP869
[14] B. Gess and M. Hofmanová. Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE. Ann. Probab.46 (5) (2018) 2495-2544. · Zbl 1428.60090 · doi:10.1214/17-AOP1231
[15] B. Gess and M. Röckner. Singular-degenerate multivalued stochastic fast diffusion equations. SIAM J. Math. Anal.47 (5) (2015) 4058-4090. · Zbl 1330.60080 · doi:10.1137/151003726
[16] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics24. Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[17] E. P. Hsu, Y. Wang and Z. Wang. Stochastic De Giorgi iteration and regularity of stochastic partial differential equations. Ann. Probab.45 (5) (2017) 2855-2866. · Zbl 1386.60219 · doi:10.1214/16-AOP1126
[18] N. V. Krylov. A relatively short proof of Itô’s formula for SPDEs and its applications. Stoch. Partial Differ. Equ. Anal. Comput.1 (1) (2013) 152-174. · Zbl 1274.60204 · doi:10.1007/s40072-013-0003-5
[19] N. V. Krylov and B. L. Rozovskii. Stochastic evolution equations. In Stochastic Differential Equations: Theory and Applications 1-69. Interdiscip. Math. Sci.2. World Sci. Publ., Hackensack, NJ, 2007. · Zbl 1130.60069
[20] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs23. American Mathematical Society, Providence, RI, 1968 (Russian). Translated from the Russian by S. Smith.
[21] J. Moser. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math.17 (1964) 101-134. · Zbl 0149.06902 · doi:10.1002/cpa.3160170106
[22] E. Pardoux. Sur des équations aux dérivées partielles stochastiques monotones. C. R. Acad. Sci. Paris Sér. A-B275 (1972) A101-A103 (French). · Zbl 0236.60039
[23] C. Prévôt and M. Röckner. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics1905. Springer, Berlin, 2007. · Zbl 1123.60001
[24] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften293. Springer-Verlag, Berlin, 1999. · Zbl 0917.60006
[25] B. L. Rozovskiǐ. Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Mathematics and Its Applications (Soviet Series)35. Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian by A. Yarkho. · Zbl 0724.60070
[26] P. E. Sacks. The initial and boundary value problem for a class of degenerate parabolic equations. Comm. Partial Differential Equations8 (7) (1983) 693-733. · Zbl 0529.35038 · doi:10.1080/03605308308820283
[27] J. L. Vázquez. The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. · Zbl 1107.35003
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