Penalized Langevin dynamics with vanishing penalty for smooth and log-concave targets

We study the problem of sampling from a probability distribution on $\mathbb R^p$ defined via a convex and smooth potential function. We first consider a continuous-time diffusion-type process, termed Penalized Langevin dynamics (PLD), the drift of which is the negative gradient of the potential plus a linear penalty that vanishes when time goes to infinity. An upper bound on the Wasserstein-2 distance between the distribution of the PLD at time $t$ and the target is established. This upper bound highlights the influence of the speed of decay of the penalty on the accuracy of approximation. As a consequence, in the case of low-temperature limit we infer a new result on the convergence of the penalized gradient flow for the optimization problem.