Geodesic
Weak and generalized with random variable solutions of stochastic Сauchy problem with additive white noise
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 1, pp. 19-27

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The article describes the solutions of an abstract stochastic Cauchy problem for the $X'(t) = AX(t)+BW(t)$ equation with the $A$ operator, which is the generator of a semigroup of $C_0$ class in a Hilbert space $H$ with the white noise $W$ in a different Hilbert space $\mathrm{H}$ and a linear operator $\mathrm{B: H}\to H$. Two approaches to solve the problem are considered: the Ito integral approach, when the integral problem is solved with ito integral following Wiener process; the approach based on the analysis of the white noise in the original differential problem in the function spaces generalized with random variable. The relation between the solutions is defined.
Keywords: stochastic Cauchy problem; white noise; Wiener process; weak solution; distribution; generalized solution. 
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     author = {O. S. Starkova},
     title = {Weak and generalized with random variable solutions of stochastic {{\CYRS}auchy} problem with additive white noise},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
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     volume = {8},
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     url = {http://geodesic.mathdoc.fr/item/VYURM_2016_8_1_a2/}
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O. S. Starkova. Weak and generalized with random variable solutions of stochastic Сauchy problem with additive white noise. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 1, pp. 19-27. http://geodesic.mathdoc.fr/item/VYURM_2016_8_1_a2/