Forward Kolmogorov equation

This equation states that the particle density at time n + 1 is the sum of the densities at intermediate points x - z at time n multiplied by the probability of transition from X - z to X. This is a mesoscopic description. Although it only deals with the mean density of particles p(x,n), it involves a detailed description of the movement of particles on the microscopic level. Equation (3.13) is the same as the Kolmogorov forward equation (3.12). The solution to (3.13) can be rewritten as aconvolution... 

The mesoscopic density is governed by the Kolmogorov forward equation, the Master equation,... 

Fokker-Planck equation The Langevin equation describes the Brownian motion of a single particle which experiences a random force (due to collisions with the solvent particles) causing the velocity to behave in a stochastic way. The Fokker-Planck equation (also known as Kolmogorov forward equation) extends the Langevin equation to an ensemble of identical Brownian particles by finding the probability distribution P v, t) of N particles in the ensemble having velocities in the interval v, V -t- St) at time t. The Fokker-Planck equation can be formally expressed as... 

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 ... 

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... 

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

Substitution of Eq.(2-112) into Eq.(2-11 la) and approaching At to zero, yields a simplified version of the forward Kolmogorov differential equation for the transition Sj -> Sij-> out of ikth city. This equation is continuous in time and... 

The Kolmogorov equation. In the preceding model, the major assumption made was that the one-step transition probabilities p and q remain constant. In the following it is assumed that these probabilities are state depending, namely, designated as pk and qk. The following forward equation for the present model reads ... 

The forward equation The Fokker-Planck equation is now derived as the differential form of the Chapman-Kolmogorov equation For any function f (x)... 

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 this section we remind the reader of the Kolmogorov forward and backward equations, infinitesimal generators, stochastic differential equations, and functional integrals and then consider how the basic transport equations are related to underlying Markov stochastic processes [141, 142],... 

Our goal is to derive the Kolmogorov forward and backward equations and to discuss the main difference between them. The forward equation deals with the events during the small time interval (t, t+h] and gives us the answer for how those events define the probability density p(y, t+h x) at time while the backward equation is concerned with events just after the time t = 0. 

The forward Kolmogorov equation (5.10) is referred to in physical literature as the Fokker-Planck equation. For the absolute density function g(x, t) (which contains less information than/) the following Kolmogorov-like equation holds ... 

Let P x, t) be the matrix of transition probabilities Pj,(r, /). It can be shown that the transition probabilities satisfy the Kolmogorov forward differential equations that is,... 

The transition probabihty column vector P(t) = Py(t), j = 1,2, 3, satisfies the Kolmogorov forward differential equation [3] ... 

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... 

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],... 

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

Note the differences between the form of the diffusion term that appears in this Stratonovich form of the diffusion equation and those that appear in the Ito (or forward Kolmogorov) form of Eq. (2.222) and in the physical diffusion equation of Eq. (2.78). 

These are the so-called forward and backward Kolmogorov equations for the Ornstein-Uhlenbeck process. Their paramount importance will appear in VIII.4 under the more familiar name of Fokker-Planck equation. 

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]. 

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