Abstract. The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in the square-mean sense is discussed.

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We present a method to learn mean residence time and escape probability from data modeled by stochastic differential equations. This method is a combination of machine learning from data (to extract stochastic differential equations as models) and stochastic dynamics (to quantify dynamical behaviors with deterministic tools). The goal is to learn and understand stochastic dynamics based on data. This method is applicable to sample path data collected from complex systems, as long as these systems can be modeled as stochastic differential equations.

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