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stochastic distributions are used to investigate existence of strong solutions of a stochastic equation with multiplicative noise on a separable Hilbert space.
The Cauchy problem for the equation u (t) = Au(t) + BW(t) , t ≥ 0, with white noise W and A being the generator of regularized semigroups is studied in different spaces of distributions. Solutions of the problem in spaces of distributions with respect to time variable, random variable and both time and random variables are studied.
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The Cauchy problem for systems of differential equations with multiplicative random perturbations in the form of infinite-dimensional Ito integrals is studied. For the systems correct by Petro-vskii, conditionally correct and incorrect we point out Gelfand–Shilov spaces of generalized functions where a generalized solution coincides with a mild solution.