Approximation of Markov semigroups in total variation distance.

Summary The first goal of this paper is to prove that, regularization properties of a Markov semigroup enable to prove convergence in total variation distance for approximation schemes for the semigroup. Moreover, using an interpolation argument we obtain estimates for the error in distribution sense (at the level of the densities of the semigroup with respect to the Lebesgue measure). In a second step, we build an abstract Malliavin calculus based on a splitting procedure, which turns out to be the suited instrument in order to prove the above mentioned regularization properties. Finally, we use these results in order to estimate the error in total variation distance for the Ninomiya Victoir scheme (which is an approximation scheme, of order 2, for diffusion processes).