Geodesic
Connection between weak and generalized solutions of infinite-dimensional stochastic problems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2014), pp. 12-27.

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We investigate the stochastic Cauchy problem for the first order equation with singular white noise and generators of regularized (integrated, convoluted) semigroups in Hilbert spaces and abstract distribution spaces. Weak solutions for the problem in the Ito form and generalized solutions for the differential problem in abstract distribution spaces are constructed in dependence on properties of the generator. Connections between these solutions are shown.
Mots-clés : distribution
Keywords: semigroup of operators, white noise, Wiener process, generalized solution, weak solution, regularized solution. 
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I. V. Mel'nikova; O. S. Starkova. Connection between weak and generalized solutions of infinite-dimensional stochastic problems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2014), pp. 12-27. http://geodesic.mathdoc.fr/item/IVM_2014_5_a1/

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

[2] Da Prato G., Kolmogorov equations for stochastic PDEs, Birkhäuser Verlag, Basel, 2004 | MR | Zbl

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

[4] Melnikova I. V., Filinkov A. I., “Abstract stochastic problems with generators of regularized semigroups”, J. Commun. Appl. Anal., 13:2 (2009), 195–212 | MR | Zbl

[5] Alshanskii M. A., Melnikova I. V., “Regulyarizovannye i obobschennye resheniya beskonechnomernykh stokhasticheskikh zadach”, Matem. sb., 202:11 (2011), 3–30 | DOI | MR | Zbl

[6] Anufrieva U. A., Melnikova I. V., “Pecularities and regularization of ill-posed Cauchy problems with differential operators”, J. Math. Sci. (New York), 148:4 (2008), 481–632 | DOI | MR | Zbl

[7] Chioranescu I., “Local convoluted semigroups”, Lecture Notes in Pure and Appl. Math., 168, 1995, 107–122 | MR

[8] Melnikova I. V., Filinkov A., Abstract Cauchy problems: three approaches, Chapman Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 120, Chapman Hall/CRC, Boca Roton, FL, 2001 | DOI | MR | Zbl

[9] Oksendal B., Stokhasticheskie differentsialnye uravneniya. Vvedenie v teoriyu i prilozheniya, Mir, M., 2003

[10] Fattorini H. O., The Cauchy problem, Addison-Wesley, Reading, Mass., 1983 | MR