  We study the problem of sampling from a probability distribution on $\mathbb
R^p$ defined via a convex and smooth potential function. We 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 the approximation. As a consequence,
considering the low-temperature limit we infer a new nonasymptotic guarantee of
convergence of the penalized gradient flow for the optimization problem.
