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Importance sampling, Fokker-Planck and Langevin equations

The transition from a state j to a state i is now replaced by a transition to a state with position \mathbf{y} from a state with position \mathbf{x} . The discrete sum of transition probabilities can then be replaced by an integral and we obtain the new distribution at a time t+\Delta t as

w(\mathbf{y},t+\Delta t)= \int W(\mathbf{y},t+\Delta t| \mathbf{x},t)w(\mathbf{x},t)d\mathbf{x},

and after m time steps we have

w(\mathbf{y},t+m\Delta t)= \int W(\mathbf{y},t+m\Delta t| \mathbf{x},t)w(\mathbf{x},t)d\mathbf{x}.

When equilibrium is reached we have

w(\mathbf{y})= \int W(\mathbf{y}|\mathbf{x}, t)w(\mathbf{x})d\mathbf{x},

that is no time-dependence. Note our change of notation for W