SPDEs in $L_q( ( 0,\tau ] , L_p)$ Spaces

N. V. Krylov (University of Minnesota)


Existence and uniqueness theorems are presented for evolutional stochastic partial differential equations of second order in $L_p$-spaces with weights allowing derivatives of solutions to blow up near the boundary. It is allowed for the powers of summability with respect to space and time variables to be different.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-29

Publication Date: June 30, 2000

DOI: 10.1214/EJP.v5-69


  1. E. B. Fabes, M. V. Safonov, and Yu Yuan, Behavior near the boundary of positive solutions of second order parabolic equations. II, Trans. Amer. Math. Soc., 351, No. 12 (1999), 4947-4961. Math. Review 2000c:35085
  2. N. V. Krylov, Nonlinear elliptic and parabolic equations of second order,Nauka, Moscow, 1985 in Russian; English translation by Reidel, Dordrecht, 1987. Math. Review 88d:35005
  3. N. V. Krylov, One-dimensional SPDEs with constant coefficients on the positive half axis, Proceedings of Steklov Mathematical Institute Seminar, Statistics and Control of Stochastic Processes, The Liptser Festschrift, Kabanov, Rozovskii, Shiryaev eds., World Scientific, Singapore-New Jersey-London-HongKong, 1997, pp. 243--251. Math. Review 99j:60092
  4. N. V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, 64, AMS, Providence, RI, 1999. Math. Review 99j:60093
  5. N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space,Comm in PDE, 24, No. 9-10 (1999), 1611-1653. Math. Review 1708104
  6. N. V. Krylov, Some properties of weighted Sobolev spaces in $bR^{d}_{+}$, Annali Scuola Normale Superiore di Pisa, Sci. Fis. Mat., Serie 4, 28 (1999), Fasc. 4, 675-693.
  7. N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, submitted to Journal of Functional Analysis.
  8. N. V. Krylov, The heat equation in $L_{q}((0,T),L_{p})$-spaces with weights,submitted to SIAM J. on Math. Anal.
  9. N. V. Krylov, S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30, no. 2 (1999) 298 - 325. Math. Review 99k:60164
  10. N. V. Krylov, S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space,SIAM J. on Math. Anal., 31, No 1 (1999), 19-33. Math. Review 1 720 129
  11. S. B. Kuksin, N. S. Nadirashvili, and A. L. Piatnitski, H"older estimates for solutions of parabolic SPDEs, preprint.
  12. S. V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains, pp. 185-242 in Stochastic and Stoch. Rep., 68 (1999), 145-175. Math. Review 1 742 721
  13. B. L. Rozovskii, Stochastic evolution systems, Kluwer, Dordrecht, 1990. Math. Review 92k:60136
  14. P. Weidemaier, On the sharp initial trace of functions with derivatives in $L_{p}(0,T;L_{p}(Omega))$, Bollettino U.M.I. (7), 9-B (1995), 321-338. Math. Review 96d:46042

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.