Diffusions for Global Optimization

We seek a global minimum of U:[0,1]ⁿ → 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 π T (x)α exp \ - 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.