Abstract Cauchy Problem in spaces of stochastic distributions
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 2, Tome 16 (2006), pp. 96-109.

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The first order abstract Cauchy problem with additive white noise and generator of an $R$-semigroup in a Hilbert space $H$ is investigated. A generalized solution in spaces of $H$-valued stochastic distributions is constructed and main characteristics of the solution are obtained.
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I. V. Mel'nikova; A. I. Filinkov. Abstract Cauchy Problem in spaces of stochastic distributions. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 2, Tome 16 (2006), pp. 96-109. http://geodesic.mathdoc.fr/item/CMFD_2006_16_a6/

[1] Melnikova I. V., Filinkov A., “Slaboe i obobschennoe resheniya abstraktnykh stokhasticheskikh uravnenii”, Dokl. AN, 375:4 (2000), 443–447 | MR

[2] Da Prato G., Stochastic evolution equations by semigroup methods, Center de Recerca Matematica, Barcelona, 1997

[3] Da Prato G., Zabczyk J., Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[4] Davies E. B., Pang M. M., “The Cauchy problem and a generalization of the Hille–Yosida theorem”, Proc. London Math. Soc., 55:1 (1987), 181–208 | DOI | MR | Zbl

[5] Filinkov A., Sorensen J., “Differential equations in spaces of abstract stochastic distributions”, Stochastics and Stochastics Reports, 72:3–4 (2002), 129–173 | DOI | MR | Zbl

[6] Hida T., Kuo H. H., Pothoff J., Streit L., White noise, Kluwer Academic, Dodrecht/Norwell, 1993 | MR | Zbl

[7] Hille E., Phillips R. S., “Functional analysis and semigroups”, Amer. Math. Soc. Colloquium. Publ., 31, 1957 | MR | Zbl

[8] Holden H., Oksendal B., Uboe J., Zhang T., Stochastic partial differential equations. A modelling, white noise functional approach., Birkhauser, Boston, Basel, Berlin, 1996 | MR | Zbl

[9] Kuo H. H., “Lectures on white noise analysis”, Soochow J. Math., 18:3 (1992), 229–300 | MR | Zbl

[10] Melnikova I. V., Anufrieva U. A., “Pecularities and regularization of ill-posed Cauchy problems with differential operators”, J. Math. Sci. (to appear)

[11] Melnikova I. V., Filinkov A., “Abstract Cauchy problem: three approaches”, Monogr. Surv. Pure Appl. Math., 120, 2001 | MR | Zbl

[12] Melnikova I. V., Filinkov A. I., Alshansky M. A., “Abstract stochastic equations II. Solutions in spaces of abstract stochastic distributions”, J. Math. Sci., 116:5 (2003), 3620–3656 | DOI | MR | Zbl

[13] Melnikova I. V., Filinkov A. I., Anufrieva U. A., “Abstract stochastic equations I. Classical and generalized solutions”, J. Math. Sci., 111:2 (2002), 3430–3475 | DOI | MR | Zbl

[14] Tanaka N., Okazawa N., “Local $C$-semigroups and local integrated semigroups”, Proc. London Math. Soc., 61:3 (1990), 63–90 | DOI | MR | Zbl

[15] Zhang T., “Characterizations of white noise test functions and Hida distributions”, Stochastics, 41 (1992), 71–87 | MR | Zbl