Annealing diffusions in a potential function with a slow growth

Abstract

Consider a continuous analogue of the simulated annealing algorithm in R d , namely the solution of the SDE d X t = σ ( t ) d B t − ∇ V ( X t ) d t , where V is a function called the potential. We prove a convergence result, similar to the one in [L. Miclo, Thèse de doctorat, Ph.D. Thesis, Université Paris VI, 1991], under weaker hypotheses on the potential function. In particular, we cover cases where the gradient of the potential goes to zero at infinity. The main idea is to replace the Poincaré and log-Sobolev inequalities used in [L. Miclo, Thèse de doctorat, Ph.D. Thesis, Université Paris VI, 1991; C.-R. Hwang, T.-S. Chiang, S.-J. Sheu, Diffusion for global optimization in Rn, SIAM J. Control Optim. 25 (1987) 737–753.] by the weak Poincaré inequalities (introduced in [M. Röckner, F.-Y. Wang, Weak Poincaré inequalities and L 2 convergence rates of Markov semigroups, J. Funct. Anal. 185 (2001) 564–603]), and to estimate constants with measure–capacity criteria. We show that the convergence still holds for the ‘classical’ schedule σ ( t ) = c / ln ( t ) , where c is bigger than a constant related to V (namely the height of the largest potential barrier).