Recuit simulésans potentiel sur une variétériemannienne compacte

Abstract

We consider general simulated annealing algorithms on a connected and compact Riemannian manifold, i.e. algorithms for which the drift is not necessarily the gradient of a given potential. When this is satisfied (or equivalently, when the process considered is instantaneously reversible), Holley, Kusuoka and Stroock proved the convergence of the algorithm, for certain decreasing rate of the temperature. The method presented here is different and is very general (it can be adapted to the situation where the phase space is an infinite dimensional one on which we are given a translation invariant and finite range family of potentials). It is based on the study of the evolution of the entropy of the law of the diffusion at one instant with respect to the instantaneous invariant probability, but we also depend on the Sobolev logarithmic inequalities which can be easily deduced from the results of the previous authors. These inequalities give us a differential inequality for the entropy, which implies that it converges to zero under decreasing rates of the temperature of the kind k/ln(t),for k strictly larger than one certain constant c ≥0. We can then prove that the stochastic process considered converge to the global minima of the quasi-potential introduced by Wentzell and Freidlin as the action functional for the large deviation principle satisfied by the invariant measure. But apart from the case where the manifold is the circle or the instantaneously reversible situation (for which there exists an explicit formula for the invariant probability), the constant is twice the natural generalisation of the constant which appear when the drift derive from a potential, so the constant c obtained is certainly not optimal.