The Annals of Probability

Wiener chaos solutions of linear stochastic evolution equations

S. V. Lototsky and B. L. Rozovskii

Full-text: Open access

Abstract

A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.

Article information

Source
Ann. Probab., Volume 34, Number 2 (2006), 638-662.

Dates
First available in Project Euclid: 9 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1147179985

Digital Object Identifier
doi:10.1214/009117905000000738

Mathematical Reviews number (MathSciNet)
MR2223954

Zentralblatt MATH identifier
1100.60034

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60H40: White noise theory

Keywords
Feynmann–Kac formula generalized random elements stochastic parabolic equations turbulent transport white noise

Citation

Lototsky, S. V.; Rozovskii, B. L. Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. 34 (2006), no. 2, 638--662. doi:10.1214/009117905000000738. https://projecteuclid.org/euclid.aop/1147179985


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References

  • Baxendale, P. and Harris, T. E. (1986). Isotropic stochastic flows. Ann. Probab. 14 1155--1179.
  • Bennett, C. and Sharpley, R. (1988). Interpolation of Operators. Academic Press, Boston.
  • Budhiraja, A. and Kallianpur, G. (1996). Approximations to the solution of the Zakai equations using multiple Wiener and Stratonovich integral expansions. Stochastics Stochastics Rep. 56 271--315.
  • Cameron, R. H. and Martin, W. T. (1947). The orthogonal development of nonlinear functionals in a series of Fourier--Hermite functions. Ann. of Math. 48 385--392.
  • Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
  • Deck, T. and Potthoff, J. (1998). On a class of stochastic partial differential equations related to turbulent transport. Probab. Theory Related Fields 111 101--122.
  • Doering, C. R. and Gibbon, J. D. (1995). Applied Analysis of the Navier--Stokes Equations. Cambridge Univ. Press.
  • E, W. and Vanden Eijden, E. (2000). Generalized flows, intrinsic stochasticity, and turbulent transport. Proc. Natl. Acad. Sci. USA 97 8200--8205.
  • Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Princeton Univ. Press.
  • Gawędzki, K. and Vergassola, M. (2000). Phase transition in the passive scalar advection. Phys. D 138 63--90.
  • Hida, T., Kuo, H.-H., Potthoff, J. and Sreit, L. (1993). White Noise. Kluwer Academic Publishers, Boston.
  • Holden, H., Øksendal, B., Ubøe, J. and Zhang, T. (1996). Stochastic Partial Differential Equations. Birkhäuser, Boston.
  • Itô, K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3 157--169.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Köthe, G. (1968). Über nukleare Folgenräume. Studia Math. 31 267--271. (In German.)
  • Köthe, G. (1970). Stark nukleare Folgenräume. J. Fac. Sci. Univ. Tokyo Sect. I 17 291--296. (In German.)
  • Köthe, G. (1971). Nuclear sequence spaces. Math. Balkanica 1 144--146.
  • Kraichnan, R. H. (1968). Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11 945--963.
  • Krylov, N. V. (1999). An analytic approach to SPDEs. In Stochastic Partial Differential Equations: Six Perspectives. Mathematical Surveys and Monographs (B. L. Rozovskii and R. Carmona, eds.) 185--242. Amer. Math. Soc., Providence, RI.
  • Krylov, N. V. and Veretennikov, A. J. (1976). On explicit formula for solutions of stochastic equations. Math. USSR Sbornik 29 239--256.
  • Krylov, N. V. and Zvonkin, A. K. (1981). On strong solutions of stochastic differential equations. Sel. Math. Sov. 1 19--61.
  • Kunita, H. (1982). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press.
  • LeJan, Y. and Raimond, O. (2002). Integration of Brownian vector fields. Ann. Probab. 30 826--873.
  • Lototsky, S. V., Mikulevicius, R. and Rozovskii, B. L. (1997). Nonlinear filtering revisited: A spectral approach. SIAM J. Control. Optim. 35 435--461.
  • Lototsky, S. V. and Rozovskii, B. L. (2004). Passive scalar equation in a turbulent incompressible Gaussian velocity field. Rusian Math. Surveys 59 297--312.
  • Matingly, J. C. and Sinai, Y. G. (1999). An elementary proof of the existence and uniqueness theorem for the Navier--Stokes equations. Commun. Contemp. Math. 1 497--516.
  • Mikulevicius, R. and Rozovskii, B. L. (1993). Separation of observations and parameters in nonlinear filtering. In Proc. 32nd IEEE Conf. on Decision and Control, San Antonio, Texas, 1993 2 1564--1559. IEEE Control Systems Society.
  • Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDE's and Wiener chaos. SIAM J. Math. Anal. 29 452--480.
  • Mikulevicius, R. and Rozovskii, B. L. (2001). Stochastic Navier--Stokes equations. Propagation of chaos and statistical moments. In Optimal Control and Partial Differential Equations: In Honour of Alain Bensoussan (J. L. Menaldi, E. Rofman and A. Sulem, eds.) 258--267. IOS Press, Amsterdam.
  • Nualart, D. and Rozovskii, B. (1997). Weighted stochastic Sobolev spaces and bilinear SPDE's driven by space--time white noise. J. Funct. Anal. 149 200--225.
  • Ocone, D. (1983). Multiple integral expansions for nonlinear filtering. Stochastics 10 1--30.
  • Øksendal, B. K. (1998). Stochastic Differential Equations: An Introduction With Applications, 5th ed. Springer, Berlin.
  • Pardoux, E. (1975). Equations aux derivéespartielles stochastiques non linearies monotones. Etude de solutions fortes de type Itô. Univ. Paris Sud, thése Doct. Sci. Math.
  • Potthoff, J., Våge, G. and Watanabe, H. (1998). Generalized solutions of linear parabolic stochastic partial differential equations. Appl. Math. Optim. 38 95--107.
  • Rozovskii, B. L. (1990). Stochastic Evolution Systems. Kluwer Academic Publishers, Dordrecht.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.
  • Wong, E. (1981). Explicit solutions to a class of nonlinear filtering problems. Stochastics 16 311--321.