Approximation of Markov semigroups in total variation distance

In this paper, we consider Markov chains of the form Xⁿ_(k+1)/n = ψ_k(Xⁿ_k/n, Z_k+1/√n, 1/n) where the innovation comes from the sequence Z_k, k ∈ ℕ* of independent centered random variables with arbitrary law. Then, we study the convergence (image found)[f(Xⁿ_t)] → (image found)[f(X_t)] where (X_t)_t≥ 0 is a Markov process in continuous time. This may be considered as an invariance principle, which generalizes the classical Central Limit Theorem to Markov chains. Alternatively (and this is the main motivation of our paper), Xⁿ may be an approximation scheme used in order to compute (image found)[f(X_t)] by Monte Carlo methods. Estimates of the error are given for smooth test functions f as well as for measurable and bounded f: In order to prove convergence for measurable test functions we assume that Z_k satisfies Doeblin’s condition and we use Malliavin calculus type integration by parts formulas based on the smooth part of the law of Z_k. As an application, we will give estimates of the error in total variation distance for the Ninomiya Victoir scheme.