  In this paper, we study a method to sample from a target distribution $\pi$
over $\mathbb{R}^d$ having a positive density with respect to the Lebesgue
measure, known up to a normalisation factor. This method is based on the Euler
discretization of the overdamped Langevin stochastic differential equation
associated with $\pi$. For both constant and decreasing step sizes in the Euler
discretization, we obtain non-asymptotic bounds for the convergence to the
target distribution $\pi$ in total variation distance. A particular attention
is paid to the dependency on the dimension $d$, to demonstrate the
applicability of this method in the high dimensional setting. These bounds
improve and extend the results of (Dalalyan 2014).
