Multidimensional Stein's method for asymptotic independence with invariant measures of diffusion

We derive a multidimensional Stein's method for asymptotic independence in the case of a general target $\mu$ with a density, being invariant measure of a diffusion process. It allows us to give a general bound in Wasserstein distance between the law of a couple $(X, Y)$, where $X$ is a random variable, and $Y$ a random vector and $\mu \otimes \mathrm{Law}(Y)$. We focus in particular in the case where $X$ and $Y$ are differentiable in the Malliavin sense, by being function of a finite number of stochastic Wiener integrals.