Kolmogorov s forward equation

These differential equations depend on the entire probability density function / (x, t) for x(t). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(x, t). 

Kolmogorov s Forward Equation The transition probability p(s, x t, y) satisfies the forward equation ... 

The n-variable version of Kolmogorov s forward equation (or Fokker-Planck equation) can be written as... 

It can easily be verified that the transition probability density verifies the forward Kolmogorov s equation (4.116), with M(t, y) = 0 and jS2(t, y) = D, which is the familiar diffusion equation homogeneous in space and time... 

These are called Kolmogorov s backward and forward equations, respectively. The Chapman-Kolmogorov equations P(s + t) = P(s)P(t) may be deduced from (30) as follows ... 

In the language of applied maths, p(x, y, t) is the Green s function for the diffusion process. It is important to note that in the forward equation, differentiation is carried out with respect to y (the current position) and with respect to jc (the initial position) in the backward equation. In the simulation of chemical systems, the drift term (p.) arising in Kolmogorov s equation is normally due to the Coulombic interaction between charged species and can be expressed as... 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

---