Stochastic $\Sigma$-convergence in Orlicz setting and Applications

TL;DR

This paper extends stochastic $\Sigma$-convergence to Orlicz-Sobolev spaces, enabling analysis of coupled stochastic and deterministic homogenization problems with applications to oscillatory minimization problems.

Contribution

It introduces a new convergence concept combining $\Sigma$-convergence and stochastic two-scale convergence within Orlicz spaces, and applies it to homogenization of complex variational problems.

Findings

Developed stochastic $\Sigma$-convergence in Orlicz-Sobolev spaces.

Applied the concept to homogenize oscillatory minimization problems.

Derived concrete homogenization results under various structural hypotheses.

Abstract

This paper aims to extend the concept of stochastic $\Sigma$-convergence to the framework of Orlicz-Sobolev spaces in order to deals with coupled stochastic and deterministic homogenization problems in this type of spaces. Thus, this concept is a combination of both well-known $\Sigma$-convergence [\textit{Acta Math. Sinica, English Series} \textbf{30}(9) 1621-1654] and stochastic two-scale convergence in the mean schemes [\textit{Asympt. Anal. (2025)} \textbf{142}, 291-320]. An application to the stochastic-deterministic homogenization (in the context of ergodic $H$-supralgebra) of a class of highly oscillatory minimizations problems involving integral functionals with convex and nonstandard growth integrands is also given, and some concrete homogenization problems following varied structure hypothesis are deduce from this latter.