Backward FPE

It is analogous to the forward EPE, Eq. (7), except that the differential operators on the right-hand side act on the initial state, x, instead of the current state, x, resulting in B(x) being outside of the derivative sign. Note that the backward FPE conserves the overall probability however, it cannot be written as a local continuity equation with respect to the initial position x. 

According to Section II.D.3, the translocation probability P (x) to exit through X = L, starting from an arbitrary position x, satisfies the stationary backward FPE (39), compactly written as... 

The mean transport time obeys the corresponding backward FPE... 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

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