The Linearized Benney–Luke Mathematical Model with Additive White Noise

Abstract

In the framework of the Sobolev type equations theory the linearized Benney–Luke mathematical model is considered. In studying of the model with deterministic external signal the methods and results of the Sobolev type equations theory and degenerate groups of operators are very useful, because they help to create an efficient computational algorithm. Now, the model assumes a presence of white noise along with the deterministic external force. Since the model is represented by a degenerate system of ordinary differential equations, it is difficult to apply existing nowadays approaches such as Ito-Stratonovich–Skorohod and Melnikova–Filinkov–Alshansky in which the white noise is understood as a generalized derivative of the Wiener process. 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 the article some facts of \(p\)-sectorial 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 Benney–Luke model is considered.