### EXPONENTIAL DICHOTOMIES OF A STOCHASTIC NON-CLASSICAL EQUATION ON A TWO-DIMENSIONAL SPHERE

#### Abstract

The article discusses a stochastic analogue of the linear Oskolkov equation, which is obtained from the Oskolkov system of equations. The existence of a solution to the Oskolkov stochastic equation in spaces of differential forms defined on a two-dimensional sphere. For some values of the parameters characterizing the properties of the liquid, the existence of exponential dichotomies of solutions is proved. To solve the question of the existence and stability of solutions, this equation is considered as a special case of a linear stochastic Sobolev type equation. The Nelson -- Glicklikh derivative of the stochastic process is considered as a derivative. To visualize the results obtained, an algorithm was developed for calculating stable and unstable solutions of the Oskolkov stochastic equation in spaces of 0-forms on a two-dimensional sphere. This algorithm is implemented in the Maple environment. Graphs of solutions with exponential dichotomy are plotted in a spherical coordinate system.

#### Keywords

Sobolev type equation; stochastic equations; differential forms; exponential dichotomies

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