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One class of Sobolev type equations of higher order with additive “white noise”. (English) Zbl 1331.35406

Summary: 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-Lòve 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.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35G05 Linear higher-order PDEs
35G16 Initial-boundary value problems for linear higher-order PDEs
47D09 Operator sine and cosine functions and higher-order Cauchy problems
60H30 Applications of stochastic analysis (to PDEs, etc.)
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References:

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