In this talk, I will discuss recent works in understanding the problem of learning invariant properties of ergodic Itô diffusion from time series. Applying the perturbation theory of Markov chains, I will report some convergence results of the Markov chain induced by the approximate drift and diffusion terms of stochastic differential equations in reconstructing the invariant statistics of the underlying dynamics. When the stationary density solves a steady-state Fokker-Planck equation, we leverage this perturbation theory to understand the convergence of the deep learning of the density approximation, which involves regressions of the drift, diffusion, and PDE solutions. I will elucidate the approach with a numerical example involving a 20-dimensional Langevin dynamical system.