Simulated Annealing In $\mathbf{R}^d$ With Slowly Growing Potentials

We use a localization procedure to weaken the growth assumptions of Royer [8], Miclo [4] and Zitt [9] concerning the continuous-time simulated annealing in $\mathbf{R}^d$. We show that a transition occurs for potentials growing like $a \log \log |x|$ at infinity. We also study a class of potentials with possibly unbounded sets of local minima.