We consider in this paper the problem of sampling a probability distribution $\pi$ having a density w.r.t. the Lebesgue measure on $R^d$, known up to a normalisation factor $x \mapsto e^{-U(x)}/\int_{R^d} e^{-U(y)} dy$. Under the assumption that $U$ is continuously differentiable, $\nabla U$ is globally Lipshitz and $U$ is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distances of the sampling method based on the Euler discretization of the Langevin stochastic differential equation for both constant and decreasing step sizes.

The dependence on the dimension of the state space of the obtained bounds is studied to demonstrate the applicability of this method in the high dimensional setting. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality for Lipschitz functions are reported. Some numerical results are presented to illustrate our findings.