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The generalized well-posedness of the Cauchy problem for an abstract stochastic equation with multiplicative noise

“…Let {e j } ∞ j=1 be an orthonormal basis in the space H. The family of H-valued functions of the form {h α e j } α∈T ,j∈N forms an orthogonal basis of (L 2 )(H). Elements f ∈ (L 2 )(H) are expanded in Fourier series in this basis as follows [4][5][6]:…”

Solution of an abstract Cauchy problem with nonlinear and random perturbations in the Colombeau algebra

“…We define the white noise process W as an ω mea surable generalized (in t) function with values in ‫.ވ‬ One of the ways of such a definition is based on ideas of abstract stochastic distributions (see, e.g., [5,2]). Let = (‫)ޒ‬ be the space of rapidly decreasing func tions.…”

“…problem of determining white noise and the problem related to the unboundedness of the solution operators of the homogeneous problem have been overcome; namely, solutions generalized in the time variable, the random variable, and the variable of the space H have been constructed [2,5]. However, for the nonlinear problem (1), such an approach involves multiplying generalized functions, which are, in addition, Hilbert valued.…”

Multipoint Initial-Final Value Problem for the Model of Devis with Additive White Noise

“…We also note the approach presented by the school of I.V. Melnikova, in which equation (4) is considered in Schwartz spaces, where the generalized derivative of a Wiener Kprocess makes sense [3]. Meanwhile, a new approach is actively developing in the studies of equation (4) …”

“…Fix τ 0 = 0, τ j ∈ R + (τ j−1 < τ j ), u j ∈ U, j = 0, n, and consider the multipoint initial-nal value condition (6) for a linear Sobolev type equation (3)…”