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We compare different modes of pseudo almost automorphy and variants forstochastic processes: in probability, in quadratic mean, or in distribution invarious senses. We show by a counterexample that square-mean (pseudo) almostautomorphy is a property which is too strong for stochastic differentialequations (SDEs). Finally, we consider two semilinear SDEs, one with almostautomorphic coefficients and the second one with pseudo almost automorphiccoefficients, and we prove the existence and uniqueness of a mild solutionwhich is almost automorphic in distribution in the first case, and pseudoalmost automorphic in distribution in the second case.

Almost automorphy and various extensions for stochastic processes