  In this paper, we focus on non-asymptotic bounds related to the Euler scheme
of an ergodic diffusion with a possibly multiplicative diffusion term
(non-constant diffusion coefficient). More precisely, the objective of this
paper is to control the distance of the standard Euler scheme with decreasing
step (usually called Unajusted Langevin Algorithm in the Monte-Carlo
literature) to the invariant distribution of such an ergodic diffusion. In an
appropriate Lyapunov setting and under uniform ellipticity assumptions on the
diffusion coefficient, we establish (or improve) such bounds for Total
Variation and L 1-Wasserstein distances in both multiplicative and additive and
frameworks. These bounds rely on weak error expansions using Stochastic
Analysis adapted to decreasing step setting.
