Stochastic Hölder continuity of random fields governed by a system of stochastic PDEs. (English. French summary) Zbl 1434.60153
Summary: 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.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35K45 | Initial value problems for second-order parabolic systems |
Keywords:
stochastic partial differential system; stochastic parabolicity condition; Schauder estimate; stochastic continuityReferences:
| [1] | R. A. Adams and J. J. F. Fournier. Sobolev Spaces, 2nd edition. Pure and Applied Mathematics (Amsterdam) 140, xiv+305. Elsevier/Academic Press, Amsterdam, 2003. |
| [2] | V. Bally, A. Millet and M. Sanz-Solé. Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23 (1) (1995) 178-222. · Zbl 0835.60053 |
| [3] | Z. Brzeźniak, B. Goldys and T. Jegaraj. Weak solutions of a stochastic Landau-Lifshitz-Gilbert equation. Appl. Math. Res. Express. AMRX 2013 (1) (2013) 1-33. · Zbl 1272.60041 |
| [4] | Z. Brzeźniak and M. Veraar. Is the stochastic parabolicity condition dependent on \(p\) and \(q\)? Electron. J. Probab. 17 (56) (2012) 1-24. · Zbl 1266.60113 |
| [5] | Z. Q. Chen and K. H. Kim. An \(L^p\)-theory for non-divergence form SPDEs driven by Lévy processes. Forum Math. 26 (2014) 1381-1411. · Zbl 1296.60163 · doi:10.1515/forum-2011-0123 |
| [6] | P. L. Chow and J.-L. Jiang. Stochastic partial differential equations in Hölder spaces. Probab. Theory Related Fields 99 (1) (1994) 1-27. · Zbl 0799.60053 |
| [7] | G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions, 2nd edition. Encyclopedia of Mathematics and Its Applications 152, xviii+493. Cambridge University Press, Cambridge, 2014. · Zbl 1317.60077 |
| [8] | R. Dalang, D. Khoshnevisan and E. Nualart. Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 231-271. · Zbl 1170.60322 |
| [9] | H. Dong and H. Zhang. Schauder estimates for higher-order parabolic systems with time irregular coefficients. Calc. Var. Partial Differential Equations 54 (1) (2015) 47-74. · Zbl 1323.35073 · doi:10.1007/s00526-014-0777-y |
| [10] | K. Du and J. Liu. A Schauder estimate for stochastic PDEs. C. R. Math. Acad. Sci. Paris 354 (4) (2016) 371-375. · Zbl 1387.35636 · doi:10.1016/j.crma.2016.01.010 |
| [11] | K. Du and J. Liu. On the Cauchy problem for stochastic parabolic equations in Hölder spaces. Trans. Amer. Math. Soc. 371 (4) (2019) 2643-2664. · Zbl 1404.35493 · doi:10.1090/tran/7533 |
| [12] | F. Flandoli. Dirichlet boundary value problem for stochastic parabolic equations: Compatibility relations and regularity of solutions. Stochastics 29 (3) (1990) 331-357. · Zbl 0696.60057 · doi:10.1080/17442509008833620 |
| [13] | T. Funaki. Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89 (1983) 129-193. · Zbl 0531.60095 · doi:10.1017/S0027763000020298 |
| [14] | M. Giaquinta. Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhauser, 1993. · Zbl 0786.35001 |
| [15] | D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. · Zbl 1042.35002 |
| [16] | M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) (2006) 993-1032. · Zbl 1130.37038 · doi:10.4007/annals.2006.164.993 |
| [17] | E. Hille and R. S. Phillips. Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications 31. American Mathematical Society, Providence, RI, 1957. · Zbl 0078.10004 |
| [18] | K. H. Kim and K. Lee. A \({W}^n_2\)-theory of stochastic parabolic partial differential systems on \({C}^1\)-domains. Potential Anal. 38 (3) (2013) 951-984. · Zbl 1266.60118 · doi:10.1007/s11118-012-9302-0 |
| [19] | K. H. Kim and K. Lee. A note on \(W^{\gamma}_p\)-theory of linear stochastic parabolic partial differential systems. Stochastic Process. Appl. 123 (1) (2013) 76-90. · Zbl 1256.60021 |
| [20] | N. V. Krylov. On \(L_p\)-theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27 (2) (1996) 313-340. · Zbl 0846.60061 · doi:10.1137/S0036141094263317 |
| [21] | N. V. Krylov. On SPDE’s and superdiffusions. Ann. Probab. 25 (4) (1997) 1789-1809. · Zbl 0904.60047 · doi:10.1214/aop/1023481111 |
| [22] | N. V. Krylov. An analytic approach to SPDEs. In Stochastic Partial Differential Equations: Six Perspectives 185-242. Math. Surveys Monogr. 64. Amer. Math. Soc., Providence, RI, 1999. · Zbl 0933.60073 |
| [23] | N. V. Krylov and B. L. Rozovsky. On the Cauchy problem for linear stochastic partial differential equations. Izv. Ross. Akad. Nauk Ser. Mat. 41 (6) (1977) 1267-1284. · Zbl 0396.60058 · doi:10.1070/IM1977v011n06ABEH001768 |
| [24] | G. M. Lieberman. Second Order Parabolic Differential Equations, 68. World Scientific, 1996. · Zbl 0884.35001 |
| [25] | R. Mikulevicius. On the Cauchy problem for parabolic SPDEs in Hölder classes. Ann. Probab. 28 (1) (2000) 74-103. · Zbl 1044.60050 · doi:10.1214/aop/1019160112 |
| [26] | R. Mikulevicius and H. Pragarauskas. On Cauchy-Dirichlet problem in half-space for parabolic SPDEs in weighted Hölder spaces. Stochastic Process. Appl. 106 (2) (2003) 185-222. · Zbl 1075.60543 · doi:10.1016/S0304-4149(03)00042-5 |
| [27] | R. Mikulevicius and H. Pragarauskas. On Cauchy-Dirichlet problem for parabolic quasilinear SPDEs. Potential Anal. 25 (1) (2006) 37-75. · Zbl 1138.60047 · doi:10.1007/s11118-005-9006-9 |
| [28] | R. Mikulevicius and B. Rozovsky. A note on Krylov’s \(L_p\)-theory for systems of SPDEs. Electron. J. Probab. 6 (12) (2001) 1-35. · Zbl 0974.60043 · doi:10.1214/EJP.v6-85 |
| [29] | R. Mikulevicius and B. Rozovsky. Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 (5) (2004) 1250-1310. · Zbl 1062.60061 · doi:10.1137/S0036141002409167 |
| [30] | R. Mikulevicius and B. Rozovsky. On unbiased stochastic Navier-Stokes equations. Probab. Theory Related Fields 154 (3-4) (2012) 787-834. · Zbl 1277.60109 · doi:10.1007/s00440-011-0384-1 |
| [31] | C. Mueller and R. Tribe. Hitting properties of a random string. Electron. J. Probab. 7 (10) (2002) 1-29. · Zbl 1010.60059 · doi:10.1214/EJP.v7-109 |
| [32] | E. Pardoux. Equations aux dérivées partielles stochastiques non linéaires monotones; Etude de solutions fortes de type Itô. PhD thesis, Univ. Paris-Sud, Orsay, 1975. |
| [33] | P. Portal and M. Veraar. Stochastic maximal regularity for rough time-dependent problems. Stoch. Partial Differ. Equ. Anal. Comput. 7 (2019) 541-597. · Zbl 1431.60062 · doi:10.1007/s40072-019-00134-w |
| [34] | B. L. Rozovsky. Stochastic partial differential equations. Mat. Sb. 96 (138) (1975) 314-341, 344. · Zbl 0326.60070 |
| [35] | W. Schlag. Schauder and \(L^p\) estimates for parabolic systems via Campanato spaces. Comm. Partial Differential Equations 21 (7-8) (1996) 1141-1175. · Zbl 0864.35023 · doi:10.1080/03605309608821221 |
| [36] | S. Tang and W. Wei. On the Cauchy problem for backward stochastic partial differential equations in Hölder spaces. Ann. Probab. 44 (1) (2016) 360-398. · Zbl 1336.60125 · doi:10.1214/14-AOP976 |
| [37] | J. van Neerven, M. Veraar and L. Weis. Stochastic maximal \(L^p\)-regularity. Ann. Probab. 40 (2) (2012) 788-812. · Zbl 1249.60147 · doi:10.1214/10-AOP626 |
| [38] | J. B. Walsh. An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV—1984 265-439. Lecture Notes in Math. 1180. Springer, Berlin, 1986. · Zbl 0608.60060 |
| [39] | M. Zakai. On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete 11 (3) (1969) 230-243. · Zbl 0164.19201 · doi:10.1007/BF00536382 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
