Convergence to global minima for a class of diffusion processes

We prove that there exists a gain function (η(t),β(t))t⩾0 such that the solution of the SDE dxt=η(t)(−gradU(xt)dt+β(t)dBt) ‘settles’ down on the set of global minima of U. In particular, the existence of a gain function (η(t))t⩾0 so that yt satisfying dyt=η(t)(−gradU(yt)dt+dBt) converges to the set of the global minima of U is verified. Then we apply the results to the Robbins–Monro and the Kiefer–Wolfowitz procedures which are of particular interest in statistics.