Diffusions for Global Optimization

We seek a global minimum of U: 0, 1ⁿ R. The solution to (d / {dt}) xₜ = - U (xₜ) will find local minima. The solution to dxₜ = - U (xₜ) dt + 2T dwₜ, where w is standard (n-dimensional) Brownian motion and the boundaries are reflecting, will concentrate near the global minima of U, at least when “temperature” T is small: the equilibrium distribution for xₜ, is Gibbs with density T (x) \ - {{U (x) / T}\}. This suggests setting T = T (t) 0, to find the global minima of U. We give conditions on U (x) and T (t) such that the solution to dxₜ = - U (xₜ) dt + 2T dwₜ converges weakly to a distribution concentrated on the global minima of U.