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We consider the Random Walk Metropolis algorithm on $\mathbb{R}^n$ withGaussian proposals, and when the target probability measure is the $n$-foldproduct 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 atequilibrium and for an appropriate scaling of the variance and of the timescaleas a function of the dimension $n$, a diffusive limit is obtained for eachcomponent of the Markov chain. In Jourdain et al. (Optimal scaling for thetransient phase of the random walk Metropolis algorithm: The mean-field limit(2012) Preprint), we generalize this result when the initial distribution isnot the target probability measure. The obtained diffusive limit is thesolution to a stochastic differential equation nonlinear in the sense ofMcKean. In the present paper, we prove convergence to equilibrium for thisequation. We discuss practical counterparts in order to optimize the varianceof the proposal distribution to accelerate convergence to equilibrium. Ouranalysis confirms the interest of the constant acceptance rate strategy (withacceptance 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 theMetropolis-Adjusted Langevin Algorithm. When starting at equilibrium, adiffusive limit for an optimal scaling of the variance is obtained in Robertsand Rosenthal (J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 (1998) 255-268). Inthe transient case, we obtain formally that the optimal variance scales verydifferently in $n$ depending on the sign of a moment of the distribution, whichvanishes at equilibrium. This suggest that it is difficult to derive practicalrecommendations for MALA from such asymptotic results.

Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior