Stepanov-like almost automorphic solutions for stochastic differential equations with Lévy noise

In this paper, we introduce a new concept of Poisson Stepanov-like almost automorphy (or Poisson S2-almost automorphy). Under some suitable conditions on the coefficients, we establish the existence and uniqueness of Stepanov-like almost automorphic mild solution to a class of semilinear stochastic differential equations with infinite dimensional Levy noise. We further discuss the global asymptotic stability of these solution. Finally, we give an example to illustrate the theoretical results obtained in this paper.