where $S_{\sigma }$ is the convergence speed-up, $S_A$ is the algorithmic speed-up, ${\sigma }_0^2$ is the variance of the original system and ${\sigma }_{ϵ}^2$ is the variance of the ARPS-Langevin system. This led to a need of a detailed analysis of the variance of ARPS-Langevin process. We performed many numerical simulations, from the simple one-dimensional case up to more real- istic dimer-solvent models, in order to observe the behavior of the variance and the quantitative dependence on the ARPS coefficients. For the one-dimensional case we managed to compute by using Galerkin approximations the numerical approximation of the variance. We are also studying analytically by use of standard techniques the properties of the ARPS-Langevin dynamics such as the existence of an invariant measure. We are also interested in the relationship between the variance of the Langevin dynamics and the ARPS-Langevin dynamics. We showed that for small ARPS coefficients the ARPS-Langevin process can be seen as a perturbation of a standard Langevin process by a perturbation operator that depends on the ARPS coefficient $ϵ$.