Linear Stochastic Parabolic Equations, Degenerating on the Boundary of a Domain

Sergey V. Lototsky (University of Southern California)


A class of linear degenerate second-order parabolic equations is considered in arbitrary domains. It is shown that these equations are solvable using special weighted Sobolev spaces in essentially the same way as the non-degenerate equations in $R^d$ are solved using the usual Sobolev spaces. The main advantages of this Sobolev-space approach are less restrictive conditions on the coefficients of the equation and near-optimal space-time regularity of the solution. Unlike previous works on degenerate equations, the results cover both classical and distribution solutions and allow the domain to be bounded or unbounded without any smoothness assumptions about the boundary. An application to nonlinear filtering of diffusion processes is discussed.

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

Pages: 1-14

Publication Date: October 17, 2001

DOI: 10.1214/EJP.v6-97


  1. Gilbarg, D. and Trudinger, N. S. (2001), Elliptic partial differential equations of second order. Reprint of the 1998 edition. Springer-Verlag, Berlin. Math. Review 2001k:35004
  2. Krylov, N. V. (1995) Introduction to the theory of diffusion processes. Translations of Mathematical Monographs, 142. American Mathematical Society, Providence, RI, Math. Review 96k:60196
  3. Krylov, N. V. (1996) Lectures on elliptic and parabolic equations in Hˆlder spaces. Graduate Studies in Mathematics, 12. American Mathematical Society, Providence, RI. Math. Review 97i:35001
  4. Krylov, N. V. (1999) An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives, 185-242, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, Math. Review 99j:60093
  5. Krylov, N. V. and Lototsky, S. V. (1999) A Sobolev space theory of SPDEs with constant coefficients on a half line. SIAM J. Math. Anal. 30, no. 2, 298-325. Math. Review 99k:60164
  6. Krylov, N. V.; Lototsky, S. V. (1999) A Sobolev space theory of SPDEs with constant coefficients in a half space. SIAM J. Math. Anal. 31, no. 1, 19-33 Math. Review 2001a:60072
  7. Lototsky, S. V. (2000) Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations. Methods Appl. Anal. 7, no. 1, 195-204. Math. Review 1 796 011
  8. Lototsky, S. V. (1999) Dirichlet problem for stochastic parabolic equations in smooth domains. Stochastics Stochastics Rep. 68, no. 1-2, 145-175. Math. Review 2000i:60069
  9. Oleinik, O. A. and Radkevic, E. V. (1973) Second order equations with nonnegative characteristic form. Plenum Press, New York-London. Math. Review 56 #16112
  10. Rozovskii, B. L. (1990) Stochastic evolution systems. Mathematics and its Applications, 35. Kluwer Academic Publishers Group, Dordrecht. Math. Review 92k:60136
  11. Shiryaev, A. N. (1996) Probability. Graduate Texts in Mathematics, 95. Springer-Verlag, New York. Math. Review 97c:60003
  12. Triebel, H. (1992) Theory of function spaces, II. Monographs in Mathematics, 84. Birkh‰user Verlag, Basel. Math. Review 93f:46029
  13. Triebel, H. (1995) Interpolation theory, function spaces, differential operators. Second edition. Johann Ambrosius Barth, Heidelberg. Math. Review 96f:46001
  14. Vishik, M. I. and Grushin, V. V. (1969) Boundary value problems for elliptic equations which are degenerate on the boundary of the domain. Math. USSR. Sb. 9, 423-454. Math. Review 41 #2212

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