  We consider the Random Walk Metropolis algorithm on $\mathbb{R}^n$ with
Gaussian proposals, and when the target probability measure is the $n$-fold
product of a one-dimensional law. It is well known (see Roberts et al. (Ann.
Appl. Probab. 7 (1997) 110-120)) that, in the limit $n\to\infty$, starting at
equilibrium and for an appropriate scaling of the variance and of the timescale
as a function of the dimension $n$, a diffusive limit is obtained for each
component of the Markov chain. In Jourdain et al. (Optimal scaling for the
transient phase of the random walk Metropolis algorithm: The mean-field limit
(2012) Preprint), we generalize this result when the initial distribution is
not the target probability measure. The obtained diffusive limit is the
solution to a stochastic differential equation nonlinear in the sense of
McKean. In the present paper, we prove convergence to equilibrium for this
equation. We discuss practical counterparts in order to optimize the variance
of the proposal distribution to accelerate convergence to equilibrium. Our
analysis confirms the interest of the constant acceptance rate strategy (with
acceptance rate between $1/4$ and $1/3$) first suggested in Roberts et al.
(Ann. Appl. Probab. 7 (1997) 110-120). We also address scaling of the
Metropolis-Adjusted Langevin Algorithm. When starting at equilibrium, a
diffusive limit for an optimal scaling of the variance is obtained in Roberts
and Rosenthal (J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 (1998) 255-268). In
the transient case, we obtain formally that the optimal variance scales very
differently in $n$ depending on the sign of a moment of the distribution, which
vanishes at equilibrium. This suggest that it is difficult to derive practical
recommendations for MALA from such asymptotic results.
