Asymptotic Stepanov automorphic properties of solutions to stochastic functional differential equations

The primary objective of this study is to identify the criteria under which a unique μ-pseudo almost automorphic solution can exist for a stochastic functional differential equation with Stepanov forcing terms. To achieve this, spectral decomposition, Ito’s isometry property and semigroup theory are utilized. The Lipschitz condition and inequality techniques are then employed to establish the uniqueness of the μ-pseudo almost automorphic solution. The methodology is demonstrated through a representative example of a stochastic process.