The Annals of Probability

On the Cauchy problem for parabolic SPDEs in Hölder classes

R. Mikulevicius

Full-text: Open access


We study Cauchy’s problem for certain second-order linear parabolic stochastic differential equation (SPDE)driven by a cylindrical Brownian motion.Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes.

Article information

Ann. Probab., Volume 28, Number 1 (2000), 74-103.

First available in Project Euclid: 18 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35K15: Initial value problems for second-order parabolic equations

Parabolic stochastic partial differential equations Cauchy problem


Mikulevicius, R. On the Cauchy problem for parabolic SPDEs in Hölder classes. Ann. Probab. 28 (2000), no. 1, 74--103. doi:10.1214/aop/1019160112.

Export citation


  • DaPrato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, New York.
  • Krylov, N. V. (1996). On Lp theory of stochastic partial differential equations. SIAM J. Math. Anal. 27 313-340.
  • Krylov, N. V. and Rozovskii, B. L. (1977). On the Cauchy problem for linear partial differential equations. Math. USSR Izvestija 11 1267-1284.
  • Ladyzhenskaja, O. A., Solonnikov, V. A. and Uraltseva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, RI.
  • Mikulevicius, R. and Pragarauskas, H. (1992). On the Cauchy problem for certain integro- differential operators in Sobolev and H¨older spaces. Lithuanian Math. J. 32 238-264.
  • Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDEs and Wiener chaos. SIAM J. Math. Anal. 29 452-480.
  • Pardoux, E. (1975). Equations aux deriv´ees partielles stochastiques non lin´eaires monotones. ´Etude de solutions de type It o, Th ese, Univ. Paris Sud, Orsay.
  • Rozovskii, B. L. (1975). On stochastic partial differential equations. Mat. Sbornik 96 314-341.
  • Rozovskii, B. L. (1990). Stochastic Evolution Systems. Kluwer, Norwell.
  • Zakai, M. (1969). On the optimal filtering of diffusion processes.Wahrsch. Verw Gebiete 11 230-243.