Abstract
The Cauchy problem for systems of differential equations with white noise type random perturbations is considered as a particular case of the first order abstract Cauchy problem with generators of R-semigroups in Hilbert spaces and with Hilbert space valued random processes. A generalized Q-white noise and cylindrical white noise are introduced as generalized derivatives of Q-Wiener and cylindrical Wiener processes in special spaces of distributions; R-semigroups generated by differential operators of the systems are defined; solutions generalized over the temporal and spacial variables are constructed in spaces of type \(\mathcal{D}'(\varPsi ')\), where topological spaces Ψ′ are chosen in dependence on singularities of solution operators to the corresponding homogeneous systems.
The paper is dedicated to the blessed memory of Alfredo Lorenzi, brilliant mathematician and bright personality
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Throughout this proof the norm ∥ ⋅ ∥ denotes the norm in \(L_{2}^{m}(\mathbb{R})\). This means that \(\|f\|^{2} =\sum _{ j=1}^{m}\int _{\mathbb{R}}\vert f_{j}(x)\vert ^{2}\,\mathit{dx} =\int _{\mathbb{R}}\left (\sum _{j=1}^{m}\vert f_{j}(x)\vert ^{2}\right )\,\mathit{dx} =\int _{\mathbb{R}}\|f(x)\|_{m}^{2}\,\mathit{dx}\), where \(\|\cdot \|_{m}\) denotes the norm of vector in \(\mathbb{C}^{m}\).
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Acknowledgements
This work is partially supported by RFBR, project 13-01-00090, and by the Program of state support of RF leading universities (agreement no. 02.A03.21.0006 from 27.08.2013).
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Melnikova, I.V., Alekseeva, U.A., Bovkun, V.A. (2014). Solutions of Stochastic Systems Generalized Over Temporal and Spatial Variables. In: Favini, A., Fragnelli, G., Mininni, R. (eds) New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer INdAM Series, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-11406-4_16
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