Smith's reduction for random processes and application for stochastic differential equations

The main focus of this work is a result related to multidimensional stochastic differential equations and one-dimensional SDEs (stochastic differential equations), using the Smith's pi-projection approach. This method is a commonly used technique in mathematics to simplify complex systems by projecting them onto lower-dimensional spaces, which can facilitate analysis and resolution. In the context of multidimensional stochastic processes (i.e., systems involving multiple random variables), extending the Smith's pi-projection approach we mention would likely involve defining a multidimensional version of this projection. This could include extending the concepts of projection and dimension reduction to multiple random variables, as well as analyzing how these projections can simplify multidimensional stochastic differential equations. As an application, we will prove the existence of almost periodic solutions to n-dimensional stochastic differential equations using the Smith projection. In fact, we will generalize the result from the article [M.-A. Boudref and A. Berboucha, Existence of almost periodic solutions of stochastic differential equations with periodic coefficients, Random Oper. Stoch. Equ. 25 2017, 1, 57-70] to dimensions n > 1 .