Geodesic
Solution of nonlinear stochastic equations with a~generator of a~semigroup discontinuous at zero
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2011), pp. 90-99.

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In this paper we study the Cauchy problem for a nonlinear stochastic equation with a generator of a semigroup discontinuous at zero. We prove the unique existence of a mild solution for some class of such semigroups. We establish a connection between weak and mild solutions for such semigroups.
Keywords: abstract stochastic Cauchy problem, operator semigroup, $Q$-Wiener process, stochastic integral. 
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A. A. Ogorodnikov. Solution of nonlinear stochastic equations with a~generator of a~semigroup discontinuous at zero. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2011), pp. 90-99. http://geodesic.mathdoc.fr/item/IVM_2011_6_a10/

[1] Shreve S. E., Stochastic calculus for finance, v. II, Continuous-time models, Springer, N.Y., 2004 | MR

[2] Walsh J. B., “An introduction to stochastic partial differential equations”, École d'ete de probabilitiés de Saint-Flour 1984, Lect. Notes in Math., 1180, Springer-Verlag, N.Y., 1986, 265–439 | MR | Zbl

[3] Da Prato G., Zabczyk J., Stochastic equations in infinite dimensions, Encycl. Math. and Appl., 44, Cambridge Univ. Press, Cambridge, 1992 | MR | Zbl

[4] Melnikova I. V., “Stokhasticheskaya zadacha Koshi v prostranstvakh abstraktnykh raspredelenii”, Izv. Uralsk. gos. un-ta, 58:11 (2008), 117–137

[5] 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

[6] Melnikova I. V., Filinkov A. I., “Generalized solutions to abstract stochastic problems”, J. Integral Transforms Spec. Func., 20:3–4 (2009), 199–206 | DOI | MR

[7] Zdobnova S. V., “Abstraktnaya stokhasticheskaya zadacha Koshi s generatorom polugruppy klassa $(1,A)$”, Voronezhsk. zimnyaya matem. shk. S. G. Kreina–2006, Tez. dokl., VorGU, Voronezh, 2006, 43–44

[8] Hille E., Phillips R. S., Functional analysis and semigroups, American Mathematical Society, Providence, 1957 | MR