Exponential Dichotomies in Barenblatt-Zheltov-Kochina Model in Spaces of Differential Forms with 'Noise'

Abstract

We investigate stability of solutions in linear stochastic Sobolev type models with the relatively bounded operator in spaces of smooth differential forms defined on smooth compact oriented Riemannian manifolds without boundary. To this end, in the space of differential forms, we use the pseudo-differential Laplace–Beltrami operator instead of the usual Laplace operator. The Cauchy condition and the Showalter–Sidorov condition are used as the initial conditions. Since “white noise” of the model is non-differentiable in the usual sense, we use the derivative of stochastic process in the sense of Nelson–Gliklikh. In order to investigate stability of solutions, we establish existence of exponential dichotomies dividing the space of solutions into stable and unstable invariant subspaces. As an example, we use a stochastic version of the Barenblatt–Zheltov–Kochina equation in the space of differential forms defined on a smooth compact oriented Riemannian manifold without boundary.