The Fokker-Planck equation yields a (the solution to the equation) transition probability given by the Green's function

$$ G(y,x,\Delta t) = \frac{1}{(4\pi D\Delta t)^{3N/2}} \exp{\left(-(y-x-D\Delta t F(x))^2/4D\Delta t\right)} $$

which in turn means that our brute force Metropolis algorithm

$$ A(y,x) = \mathrm{min}(1,q(y,x))), $$

with \( q(y,x) = |\Psi_T(y)|^2/|\Psi_T(x)|^2 \) is now replaced by the Metropolis-Hastings algorithm as well as Hasting's article,

$$ q(y,x) = \frac{G(x,y,\Delta t)|\Psi_T(y)|^2}{G(y,x,\Delta t)|\Psi_T(x)|^2} $$