Escaping a local minimum: GF vs SDE
On a double-well potential the deterministic gradient flow \dot x=-f'(x) is trapped in its starting basin. The stochastic flow dx(t) = -f'\bigl(x(t)\bigr)\,dt + \sigma\,dW(t) keeps exploring: it climbs over the barrier and its density relaxes to the Gibbs distribution \rho^*(x)\propto e^{-2f(x)/\sigma^2}, concentrated on the global minimum. The bottom bars track the fraction of trajectories that have reached the global well.