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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 y from a state with position x. The discrete sum of transition probabilities can then be replaced by an integral and we obtain the new distribution at a time t+Δt as

w(y,t+Δt)=W(y,t+Δt|x,t)w(x,t)dx,

and after m time steps we have

w(y,t+mΔt)=W(y,t+mΔt|x,t)w(x,t)dx.

When equilibrium is reached we have

w(y)=W(y|x,t)w(x)dx,

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