The integral representation immediately shows that G (r) is positive, since it is represented as a weighted sum of positive Gaussians. It also gives the rate of decay at large r, since the proper time for a random walk to reach position τ is r and in this time, the Gaussian height has decayed by e − − t τ τ = e − − t r 2 . The decay factor appropriate for position r is therefore e − − t r .