Measure Theory and Square-Mean Pseudo Almost Periodic and Automorphic Process: Application to Stochastic Evolution Equations

In this work, we establish a new concept of square-mean pseudo almost periodic and automorphic processes using the measure theory. We use the μ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }\!-$$\end{document}ergodic process to define the spaces of μ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }\!-$$\end{document}pseudo almost periodic and automorphic processes in the square-mean sense. We establish many interesting results on those spaces like completeness and composition theorems. Then, we study the existence, the uniqueness, and the stability of the square-mean μ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu }\!-$$\end{document}pseudo almost periodic and automorphic solutions of the stochastic evolution equation. We provide an example to illustrate ours results.