Weighted Stochastic Field Exponent Sobolev Spaces and Nonlinear Degenerated Elliptic Problem

In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(\mathrm{.},\mathrm{.})}(D\times \mathrm{\Omega })$ and $W_{\vartheta }^{k,p(.,.)}\left( D\times \Omega \right) $. Also, we investigate some basic properties and embeddings of these spaces. Finally, we present an application of these spaces to the stochastic partial differential equations with stochastic field growth.