Stein method for invariant measures of diffusions via Malliavin calculus

Abstract: Given a random variable $F$ regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and almost any continuous probability law on the real line. The bounds are given in terms of the Malliavin derivative of $F$. Our approach is based on the theory of It^o diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.