Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing

We study convergence rates of $\mathbb{R}$-valued algorithms, especially in the case of multiple targets and simulated annealing. We precise, for example, the convergence rate of simulated annealing algorithms, whose weak convergence to a distribution concentrated on the potential's minima had been established by Gelfand and Mitter or by Hwang and Sheu.