Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise

Abstract

Sobolev type equations theory experiences an epoch of blossoming. The majority of researches is devoted to the determined equations and systems. However in natural experiments there are the mathematical models containing accidental indignation, for example, white noise. Therefore recently even more often there arise the researches devoted to the stochastic differential equations. In the given work the Boussinesq - L$\grave{o}$ve model with additive white noise is considered within the Sobolev type equations theory. At studying the methods and results of theory of Sobolev type equations with relatively p-bounded operators were very useful. As the model is presented by the degenerate equation of mathematical physics, so it is difficult to apply existing nowadays Ito - Stratonovich - Skorokhod approaches. We use already well proved at the investigation of Sobolev type equations the phase space method consisting in a reduction of singular equation to regular one, defined on some subspace of initial space. In the first part of article some facts of $(A,p)$-bounded operators are collected. In the second - the Cauchy problem for the stochastic Sobolev type equation of high order is investigated. As an example the stochastic Boussinesq - L$\grave{o}$ve model is considered.