In this paper, we introduce a concept of Poisson $p$-mean almost automorphy for stochastic processes and give the composition theorems for (Poisson) $p$-mean almost automorphic functions under non-Lipschitz conditions. Our abstract results are, subsequently, applied to study a class of neutral stochastic evolution equations driven by L\'evy noise, and we present sufficient conditions for the existence of square-mean almost automorphic mild solutions. An example is provided to illustrate the effectiveness of the proposed result.
The project was supported by the National Natural Science
Foundation of China (Grant Nos. 11401010; 1132~6171), the Natural Science Foundation
of Anhui Province (Grant No. 1708085MA03) and the Distinguished Young Scholars
Foundation of Anhui Province (Grant No. 1608085J06).