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