The transition probability P t , t ′ ( x ∣ ∣ x ′ ) , the probability of going from ( t ′ , x ′ ) to ( t , x ) , is introduced here; the expectation can be written as E ( p ( X t + Δ Δ t ) ∣ ∣ X t = x ) = ∫ ∫ p ( y ) P t + Δ Δ t , t ( y ∣ ∣ x ) d y . Now we replace in the definition of L , multiply by P t , t ′ ( x ∣ ∣ x ′ ) and integrate over d x . The limit is taken on ∫ ∫ p ( y ) ∫ ∫ P t + Δ Δ t , t ( y ∣ ∣ x ) P t , t ′ ( x ∣ ∣ x ′ ) d x d y − − ∫ ∫ p ( x ) P t , t ′ ( x ∣ ∣ x ′ ) d x . Note now that ∫ ∫ P t + Δ Δ t , t ( y ∣ ∣ x ) P t , t ′ ( x ∣ ∣ x ′ ) d x = P t + Δ Δ t , t ′ ( y ∣ ∣ x ′ ) , which is the Chapman–Kolmogorov theorem. Changing the dummy variable y to x , one gets ∫ ∫ p ( x ) lim Δ Δ t → → 0 1 Δ Δ t ( P t + Δ Δ t , t ′ ( x ∣ ∣ x ′ ) − − P t , t ′ ( x ∣ ∣ x ′ ) ) d x , which is a time derivative. Finally we arrive to ∫ ∫ [ L p ( x ) ] P t , t ′ ( x ∣ ∣ x ′ ) d x = ∫ ∫ p ( x ) ∂ ∂ t P t , t ′ ( x ∣ ∣ x ′ ) d x . From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of L , L † † , defined such that ∫ ∫ [ L p ( x ) ] P t , t ′ ( x ∣ ∣ x ′ ) d x = ∫ ∫ p ( x ) [ L † † P t , t ′ ( x ∣ ∣ x ′ ) ] d x , then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation p ( x , t ) = P t , t ′ ( x ∣ ∣ x ′ ) , in its differential form reads L † † p ( x , t ) = ∂ ∂ t p ( x , t ) .