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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 6, Pages 1329–1347 (Mi izv2072)  

This article is cited in 27 scientific papers (total in 27 papers)

On the Cauchy problem for linear stochastic partial differential equations

N. V. Krylov, B. L. Rozovskii


Abstract: The Cauchy problem for stochastic linear partial differential equations is studied. The basic result of the paper is a uniqueness and existence theorem for the solutions of equations of this type in Sobolev spaces $L_p(\Omega;C([0,T],W_p^m))$ ($m\geqslant1$, $p\geqslant2$).
Bibliography: 12 titles.

Full text: PDF file (1591 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:6, 1267–1284

Bibliographic databases:

UDC: 519.2
MSC: 60H15
Received: 22.06.1976

Citation: N. V. Krylov, B. L. Rozovskii, “On the Cauchy problem for linear stochastic partial differential equations”, Izv. Akad. Nauk SSSR Ser. Mat., 41:6 (1977), 1329–1347; Math. USSR-Izv., 11:6 (1977), 1267–1284

Citation in format AMSBIB
\Bibitem{KryRoz77}
\by N.~V.~Krylov, B.~L.~Rozovskii
\paper On the Cauchy problem for linear stochastic partial differential equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 6
\pages 1329--1347
\mathnet{http://mi.mathnet.ru/izv2072}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=501350}
\zmath{https://zbmath.org/?q=an:0371.60076|0396.60058}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 6
\pages 1267--1284
\crossref{https://doi.org/10.1070/IM1977v011n06ABEH001768}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. V. Krylov, B. L. Rozovskii, “On conditional distributions of diffusion processes”, Math. USSR-Izv., 12:2 (1978), 336–356  mathnet  crossref  mathscinet  zmath
    2. Hiroshi Kunita, “Cauchy problem for stochastic partial differential equations arizing in nonlinear filtering theory”, Systems & Control Letters, 1:1 (1981), 37  crossref
    3. Fabien Campillo, François Le Gland, “MLE for partially observed diffusions: direct maximization vs. the em algorithm”, Stochastic Processes and their Applications, 33:2 (1989), 245  crossref
    4. X.Y. Zhou, “Remarks on optimal controls of stochastic partial differential equations”, Systems & Control Letters, 16:6 (1991), 465  crossref
    5. István Gyöngy, Nicolai V. Krylov, “On stochastic partial differential equations with Unbounded coefficients”, Potential Anal, 1:3 (1992), 233  crossref  mathscinet  zmath
    6. Xun Yu Zhou, “A duality analysis on stochastic partial differential equations”, Journal of Functional Analysis, 103:2 (1992), 275  crossref
    7. Istvàn Gyöngy, N.v. Krylov, “Stochastic partial differential equations with unbounded coefficients and applications. III”, Stochastics and Stochastic Reports, 40:1-2 (1992), 77  crossref
    8. István Gyöngy, “Stochastic partial differential equations on manifolds,I”, Potential Anal, 2:2 (1993), 101  crossref  mathscinet  zmath
    9. Hitoshi Ishii, “Viscosity solutions of nonlinear second-order partial differential equations in hilbert spaces”, Communications in Partial Differential Equations, 18:3-4 (1993), 601  crossref
    10. Xun Yu Zhou, “A class of semilinear stochastic partial differential equations and their controls: Existence results”, Stochastic Processes and their Applications, 44:1 (1993), 89  crossref
    11. R. Mikulevicius, B. Rozovskii, “Linear Parabolic Stochastic PDE and Wiener Chaos”, SIAM J Math Anal, 29:2 (1998), 452  crossref  mathscinet  zmath  isi
    12. Hyek Yoo, “Lp-estimates for stochastic PDEs with discontinuous coefficients”, Stochastic Analysis and Applications, 17:4 (1999), 687  crossref
    13. Hyek Yoo, “OnLp-Theory of stochastic partialdifferential equations of divergence form with continuous coefficients”, Stochastic Analysis and Applications, 17:5 (1999), 871  crossref
    14. R. Mikulevicius, H. Pragarauskas, “On Cauchy–Dirichlet problem in half-space for parabolic SPDEs in weighted Hölder spaces”, Stochastic Processes and their Applications, 106:2 (2003), 185  crossref
    15. Sergei Kuksin, Armen Shirikyan, “Randomly forced CGL equation: stationary measures and the inviscid limit”, J Phys A Math Gen, 37:12 (2004), 3805  crossref  mathscinet  zmath  isi  elib
    16. R. Mikulevicius, H. Pragarauskas, “On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs”, Potential Anal, 25:1 (2006), 37  crossref  mathscinet  zmath  isi
    17. R. Mikulevicius, H. Pragarauskas, N. Sonnadara, “On the Cauchy-Dirichlet Problem in the Half Space for Parabolic SPDEs in Weighted Hoelder Spaces”, Acta Appl Math, 97:1-3 (2007), 129  crossref  mathscinet  zmath  isi
    18. Thilo Meyer-Brandis, “Stochastic Feynman–Kac Equations Associated to Lévy–Itô Diffusions”, Stochastic Analysis and Applications, 25:5 (2007), 913  crossref
    19. István Gyöngy, Annie Millet, “Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations”, Potential Anal, 2008  crossref  isi
    20. R. Mikulevicius, H. Pragarauskas, “On Hölder solutions of the integro-differential Zakai equation”, Stochastic Processes and their Applications, 119:10 (2009), 3319  crossref
    21. R. Mikulevicius, H. Pragarauskas, “Model Problem for Integro-Differential Zakai Equation with Discontinuous Observation Processes”, Appl Math Optim, 2011  crossref
    22. Eric Joseph Hall, “Accelerated Spatial Approximations for Time Discretized Stochastic Partial Differential Equations”, SIAM J. Math. Anal, 44:5 (2012), 3162  crossref
    23. Kai Du, Shanjian Tang, Qi Zhang, “-solution () of linear degenerate backward stochastic partial differential equations in the whole space”, Journal of Differential Equations, 254:7 (2013), 2877  crossref
    24. E.J.oseph Hall, “Higher Order Spatial Approximations for Degenerate Parabolic Stochastic Partial Differential Equations”, SIAM J. Math. Anal, 45:4 (2013), 2071  crossref
    25. Kai Du, Shaokuan Chen, “Backward stochastic partial differential equations with quadratic growth”, Journal of Mathematical Analysis and Applications, 2014  crossref
    26. Máté Gerencsér, István Gyöngy, Nicolai Krylov, “On the solvability of degenerate stochastic partial differential equations in Sobolev spaces”, Stoch PDE: Anal Comp, 2014  crossref
    27. Du K., Liu J., “A Schauder estimate for stochastic PDEs”, C. R. Math., 354:4 (2016), 371–375  crossref  mathscinet  isi  scopus
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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