Stochastic Equations of Sobolev Type with Relatively $$p $$ -Radial Operators in Spaces of Differential Forms

We consider the Showalter–Sidorov problem for the stochastic version of the linearGinzburg–Landau equation in Hilbert spaces of smooth differential forms defined on a compactoriented Riemannian manifold without boundary with stochastic processes serving as coefficients.This equation is reduced to an abstract Sobolev type stochastic equation with a relatively radialoperator on the right-hand side, for which the solvability of the Showalter–Sidorov problem isestablished and the stability of solutions is investigated using dichotomies. Differentiation ofstochastic processes that are the coefficients of differential forms is understood in the sense of theNelson–Gliklikh derivative.