Linear Sobolev type equations with relatively \(p\)-sectorial operators in space of “noises”. (English) Zbl 1345.60054
Summary: The concept of “white noise”, initially established in finite-dimensional spaces, is transferred to the infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev-type equations and to elaborate applications of practical interest. To reach this goal, the Nelson-Gliklikh derivative is introduced and the spaces of “noises” are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable “noises”. The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition are considered as applications.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H40 | White noise theory |
| 47D99 | Groups and semigroups of linear operators, their generalizations and applications |
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