Abstract
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric hyperbolic systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brzeźniak, Z., Flandoli, F., Maurelli, M.: Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity. arXiv:1401.5938
Dalang, R.C.: Extending martingale measure stochastic integral, with applications to spatially homogeneous S.P.D.E’s. Electron. J. Probab. 4, 1–21 (1999)
Dalang, R.C., Quer-Sardanyons, L.: Stochastic integrals for spde’s: a comparison. Expositiones Mathematicae 29, 67–109 (2011)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Gerencsér, M., Gyöngy, I.: On stochastic finite difference schemes. Appl. Math. Opt. (2014). doi:10.1007/s00245-014-9272-2
Gyöngy, I., Krylov, N.V.: On stochastic equations with respect to semimartingales I. Stochastics 4(1), 1–21 (1980)
Gyöngy, I., Krylov, N.V.: Stochastic partial differential equations with unbounded coefficients and applications III. Stoch. Stoch. Rep. 40, 77–115 (1992)
Gyöngy, I., Krylov, N.V.: On the rate of convergence of splitting-up approximations for SPDEs. In Progress in Probability, vol. 56, pp. 301–321. Birkhäuser, Berlin (2003)
Gyöngy, I., Krylov, N.V.: Expansion of solutions of parametrized equations and acceleration of numerical methods. Ill. J. Math. 50, 473–514 (2006). Special Volume in Memory of Joseph Doob (1910–2004)
Gyöngy, I., Shmatkov, A.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. Appl. Math. Opt. 54(3), 315–341 (2006)
Gyöngy, I.: An introduction to the theory of stochastic partial differential equations. In preparation
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981)
Kim, K.: \(L_p\) estimates for SPDE with discontinuous coefficients in domains. Electron. J. Probab. 10, 1–20 (2005)
Kim, K.: \(L_q(L_p)\)-theory of parabolic PDEs with variable coefficients. Bull. Korean Math. Soc. 45(1), 169–190 (2008)
Kim, K., Lee, K.: A \(W^1_2\)-theory of stochastic partial differential systems of divergence type on \(C^1\) domains. Electron. J. Probab. 16, 1296–1317 (2011)
Krylov, N.V.: On SPDE’s and superdiffusions. Ann. Probab. 25, 1789–1809 (1997)
Krylov, N.V.: Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs. An analytic approach to SPDEs, pp. 185–242. American Mathematical Society, Providence, RI (1999)
Krylov, N.V.: SPDEs in \(L_q((0,\tau ], L_p)\) spaces. Electron. J. Prob. 5(13), 1–29 (2000)
Krylov, N.V.: Itô’s formula for the \(L_p\)-norm of stochastic \(W_ p^1\)-valued processes. Probab. Theory Relat. Fields 147, 583–605 (2010)
Krylov, N.V., Rozovskii, B.L.: On the Cauchy problem for linear stochastic partial differential equations. Math. USSR Izvestija 11(6), 1267–1284 (1977)
Krylov, N.V., Rosovskii, B.L.: Stochastic evolution equations. J. Soviet Math. 16, 1233–1277 (1981)
Krylov, N.V., Rozovskii, B.L.: Characteristics of degenerating second-order parabolic Itô equations. J. Soviet Maths. 32, 336–348 (1986). Translated from Trudy Seminara imeni I.G. Petrovskogo, No. 8. pp. 153–168, (1982)
Moffatt, H.K.: Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, Cambridge (1978)
Molchanov, S.A., Ruzmaikin, A.A., Sokolov, D.D.: Short-correlated random flow as a fast dynamo. Dokl. Akad. Nauk SSSR 295(3), 576–579 (1987)
Oleĭnik, O.A.: Alcuni risultati sulle equazioni lineari e quasi lineari ellittico-paraboliche a derivate parziali del secondo ordine. (Italian) Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 40(8), 775–784 (1966)
Oleĭnik, O.A.: On the smoothness of solutions of degenerating elliptic and parabolic equations. Dokl. Akad. Nauk SSSR 163, 577–580 (1965). in Russian; English translation in Soviet Mat. Dokl., Vol. 6, No. 3, 972–976 (1965)
Oleĭnik, O.A., Radkevič, E.V.: Second order equations with nonnegative characteristic form, Mathematical Analysis, 1969, pp. 7–252. (errata insert) Akad. Nauk SSSR, Vsesojuzn. Inst. Naučn. i Tehn. Informacii, Moscow, : in Russian, p. 1973. Plenum Press, New York-London, English translation (1971)
Olejnik, O.A., Radkevich, E.V.: Second Order Equations with Nonnegative Characteristic Form. AMS, Providence (1973)
Rozovskii, B.L.: Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht (1990)
Walsh, J.B.: An introduction to stochastic partial differential equations. d’Eté de Prob. de St-Flour XIV, 1984. Lecture Notes in Mathematics, vol. 1180. Springer, New York (1986)
Acknowledgments
The results of this paper were presented at the 9th International Meeting on “Stochastic Partial Differential Equations and Applications” in Levico Terme in Italy, in January, 2014, and at the meeting on “Stochastic Processes and Differential Equations in Infinite Dimensional Spaces” in King’s College London, in March, 2014. The authors would like to thank the organisers for these possibilities. The authors are grateful to the referee whose comments and suggestions helped to improve the presentation of the paper. The work of the third author was partially supported by NSF Grant DMS-1160569.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gerencsér, M., Gyöngy, I. & Krylov, N. On the solvability of degenerate stochastic partial differential equations in Sobolev spaces. Stoch PDE: Anal Comp 3, 52–83 (2015). https://doi.org/10.1007/s40072-014-0042-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40072-014-0042-6