  Using a localization procedure and the result of Holley-Kusuoka-Stroock [7]
in the torus, we widely weaken the usual growth assumptions concerning the
success of the continuous-time simulated annealing in $\mathbf{R}^d$. Our only
assumption is the existence of an invariant probability measure for a
sufficiently low temperature. We also prove, in an appendix, a non-explosion
criterion for a class of time-inhomogeneous diffusions.
