DIFFUSION FOR GLOBAL OPTIMIZATION IN RN

We seek a global minimum of U:R**n yields R. The solution to (*) (d/dt)X(t) equals minus DEL U(X(t)) will find local minima. Using the idea of simulated annealing, we consider the diffusion process, dX(t) equals minus DEL U(X(t)) dt plus nu (t) dW(T), X(0) equals x, where W( ) is the n-dimensional standard Brownian motion and one-half nu **2(t) is the annealing rate which decreases to zero as t goes to infinity . Under suitable conditions on U(x), we prove that X(t) converges weakly to a probability measure pi if for large t, nu **2(t) equals c/log t with c greater than c//0, where c//0 has a simple expression involving the action function of the dynamical system (*), pi concentrates on the global minima of U and is the weak limit of the Gibbs densities pi //t(x) varies directly as exp( minus 2U(x)/ nu **2(t)).