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Lototsky, S. V.; Rozovskii, B. L.

Wiener chaos solutions of linear stochastic evolution equations. (English) Zbl 1100.60034

Ann. Probab. 34, No. 2, 638-662 (2006).
Stochastic evolution equations may have solutions (based on weighted Wiener chaos) when the coefficients do not satisfy the ellipticity assumption, and when the underlying noise is not too regular.
Reviewer: George Stoica (Saint John)

Cited in 22 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60H40 White noise theory
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References:

[1] Baxendale, P. and Harris, T. E. (1986). Isotropic stochastic flows. Ann. Probab. 14 1155–1179. · Zbl 0606.60014 · doi:10.1214/aop/1176992360
[2] Bennett, C. and Sharpley, R. (1988). Interpolation of Operators . Academic Press, Boston. · Zbl 0647.46057
[3] 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. · Zbl 0912.60059
[4] 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. JSTOR: · Zbl 0029.14302 · doi:10.2307/1969178
[5] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions . Cambridge Univ. Press. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[6] 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. · Zbl 0906.60040 · doi:10.1007/s004400050163
[7] Doering, C. R. and Gibbon, J. D. (1995). Applied Analysis of the Navier–Stokes Equations . Cambridge Univ. Press. · Zbl 0838.76016 · doi:10.1017/CBO9780511608803
[8] E, W. and Vanden Eijden, E. (2000). Generalized flows, intrinsic stochasticity, and turbulent transport. Proc. Natl. Acad. Sci. USA 97 8200–8205. · Zbl 0967.76038 · doi:10.1073/pnas.97.15.8200
[9] Freidlin, M. (1985). Functional Integration and Partial Differential Equations . Princeton Univ. Press. · Zbl 0568.60057
[10] Gawȩdzki, K. and Vergassola, M. (2000). Phase transition in the passive scalar advection. Phys. D 138 63–90. · Zbl 0967.76041 · doi:10.1016/S0167-2789(99)00171-2
[11] Hida, T., Kuo, H.-H., Potthoff, J. and Sreit, L. (1993). White Noise . Kluwer Academic Publishers, Boston.
[12] Holden, H., Øksendal, B., Ubøe, J. and Zhang, T. (1996). Stochastic Partial Differential Equations . Birkhäuser, Boston. · Zbl 0860.60045
[13] Itô, K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3 157–169. · Zbl 0044.12202 · doi:10.2969/jmsj/00310157
[14] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Springer, New York. · Zbl 0734.60060
[15] Köthe, G. (1968). Über nukleare Folgenräume. Studia Math. 31 267–271. (In German.) · Zbl 0157.20201
[16] Köthe, G. (1970). Stark nukleare Folgenräume. J. Fac. Sci. Univ. Tokyo Sect. I 17 291–296. (In German.) · Zbl 0203.11801
[17] Köthe, G. (1971). Nuclear sequence spaces. Math. Balkanica 1 144–146. · Zbl 0219.46011
[18] Kraichnan, R. H. (1968). Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11 945–963. · Zbl 0164.28904 · doi:10.1063/1.1692063
[19] 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. · Zbl 0933.60073
[20] Krylov, N. V. and Veretennikov, A. J. (1976). On explicit formula for solutions of stochastic equations. Math. USSR Sbornik 29 239–256. · Zbl 0376.60067 · doi:10.1070/SM1976v029n02ABEH003666
[21] Krylov, N. V. and Zvonkin, A. K. (1981). On strong solutions of stochastic differential equations. Sel. Math. Sov. 1 19–61. · Zbl 0481.60062
[22] Kunita, H. (1982). Stochastic Flows and Stochastic Differential Equations . Cambridge Univ. Press. · Zbl 0743.60052
[23] LeJan, Y. and Raimond, O. (2002). Integration of Brownian vector fields. Ann. Probab. 30 826–873. · Zbl 1037.60061 · doi:10.1214/aop/1023481009
[24] Lototsky, S. V., Mikulevicius, R. and Rozovskii, B. L. (1997). Nonlinear filtering revisited: A spectral approach. SIAM J. Control. Optim. 35 435–461. · Zbl 0873.60030 · doi:10.1137/S0363012993248918
[25] Lototsky, S. V. and Rozovskii, B. L. (2004). Passive scalar equation in a turbulent incompressible Gaussian velocity field. Rusian Math. Surveys 59 297–312. · Zbl 1113.76040 · doi:10.1070/RM2004v059n02ABEH000719
[26] 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. · Zbl 0961.35112 · doi:10.1142/S0219199799000183
[27] 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.
[28] Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDE’s and Wiener chaos. SIAM J. Math. Anal. 29 452–480. · Zbl 0911.60045 · doi:10.1137/S0036141096299065
[29] 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. · Zbl 1054.60512
[30] 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. · Zbl 0894.60054 · doi:10.1006/jfan.1996.3091
[31] Ocone, D. (1983). Multiple integral expansions for nonlinear filtering. Stochastics 10 1–30. · Zbl 0517.60046 · doi:10.1080/17442508308833260
[32] Øksendal, B. K. (1998). Stochastic Differential Equations : An Introduction With Applications , 5th ed. Springer, Berlin. · Zbl 0897.60056
[33] Pardoux, E. (1975). Equations aux derivéespartielles stochastiques non linearies monotones. Etude de solutions fortes de type Itô. Univ. Paris Sud, thése Doct. Sci. Math.
[34] Potthoff, J., Våge, G. and Watanabe, H. (1998). Generalized solutions of linear parabolic stochastic partial differential equations. Appl. Math. Optim. 38 95–107. · Zbl 0908.60056 · doi:10.1007/s002459900083
[35] Rozovskii, B. L. (1990). Stochastic Evolution Systems . Kluwer Academic Publishers, Dordrecht. · Zbl 0724.60070
[36] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes . Springer, Berlin. · Zbl 0426.60069
[37] Wong, E. (1981). Explicit solutions to a class of nonlinear filtering problems. Stochastics 16 311–321. · Zbl 0475.60024 · doi:10.1080/17442508108833185
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