One Class of Sobolev Type Equations of Higher Order with Additive "White Noise"

Abstract

Sobolev type equation theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise; these models are often used in natural experiments and have recently driven a large amount of research on stochastic differential equations. A new concept of ``white noise, originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higher-order Sobolev type equation theory and provide some practical applications. The main idea is to construct ``noise spaces using the Nelson -- Gliklikh derivative. Abstract results are applied to the Boussinesq -- Love model with additive ``white noise within Sobolev type equation theory. Because of their usefulness, we mainly focus on Sobolev type equations with relatively p-bounded operators. We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.