Convergence of diffusions and their discretizations: from continuous to discrete processes and back

In this paper, we establish new quantitative convergence bounds for a class of functional autoregressive models in weighted total variation metrics. To derive this result, we show that under mild assumptions explicit minorization and Foster-Lyapunov drift conditions hold. Our bounds are then obtained adapting classical results from Markov chain theory. To illustrate our results we study the geometric ergodicity of Euler-Maruyama discretizations of diffusion with covariance matrix identity. Second, we provide a new approach to establish quantitative convergence of these diffusion processes by applying our conclusions in the discrete-time setting to a well-suited sequence of discretizations whose associated stepsizes decrease towards zero.