Stochastic equations of Sobolev type with relatively \(p \)-radial operators in spaces of differential forms. (English. Russian original) Zbl 1480.60183

Shafranov, D. E.; Kitaeva, O. G.; Sviridyuk, G. A.

Differ. Equ. 57, No. 4, 507-516 (2021); translation from Differ. Uravn. 57, No. 4, 526-535 (2021).

Summary: We consider the Showalter-Sidorov problem for the stochastic version of the linear Ginzburg-Landau equation in Hilbert spaces of smooth differential forms defined on a compact oriented Riemannian manifold without boundary with stochastic processes serving as coefficients. This equation is reduced to an abstract Sobolev type stochastic equation with a relatively radial operator on the right-hand side, for which the solvability of the Showalter-Sidorov problem is established and the stability of solutions is investigated using dichotomies. Differentiation of stochastic processes that are the coefficients of differential forms is understood in the sense of the Nelson-Gliklikh derivative.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30	Applications of stochastic analysis (to PDEs, etc.)
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

\(p \)-radial operators; Showalter-Sidorov problem

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