Some mathematical models with a relatively bounded operator and additive “white noise” in spaces of sequences

The article is devoted to the research of the class of stochastic models in mathematical physics on the basis of an abstract Sobolev type equation in Banach spaces of sequences, which are the analogues of Sobolev spaces. As operators we take polynomials with real coefficients from the analogue of the Laplace operator, and carry over the theory of linear stochastic equations of Sobolev type on the Banach spaces of sequences. The spaces of sequences of differentiable "noises" are denoted, and the existence and the uniqueness of the classical solution of Showalter–Sidorov problem for the stochastic equation of Sobolev type with a relatively bounded operator are proved. The constructed abstract scheme can be applied to the study of a wide class of stochastic models in mathematical physics, such as, for example, the Barenblatt–Zheltov–Kochina model and the Hoff model.