Global Optimization by Adapted Diffusion

In this paper, we study a diffusion stochastic dynamics with a general diffusion coefficient. The main result is that adapting the diffusion coefficient to the Hamiltonian allows to escape local wide minima and to speed up the convergence of the dynamics to the global minima. We prove the convergence of the invariant measure of the modified dynamics to a measure concentrated on the set of global minima and show how to choose a diffusion coefficient for a certain class of Hamiltonians.