  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.
