Dynamics of the compact almost automorphic solution for a class of stochastic nonlinear differential equations

This paper mainly considers the existence of the compact almost automorphic mild solution for a class of stochastic nonlinear differential equations. More specifically, based on $C_0$-semigroup theory, Hölder inequality, Burkholder–Davis–Gundy inequality and Lebesgue dominated convergence theorem, we obtain that the $K$-mild solution is uniformly continuous and is relatively compact, etc. Combined with the subvariant functional method, we give some sufficient conditions to make sure that there exists at least one minimal $K$-mild solution; further, if the minimal $K$-mild solution is unique, then it is compact and almost automorphic. Moreover, we provide an example to illustrate the main presented results.