Kolmogorov equation, probability distributions

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. 

If the incoming call distribution is Poisson, and the time to service a call has an exponential distribution (Eq. 11 with density parameter p), then the transitional probabilities related to states s0 (the server is free) and Si (the server is engaged) are Pio= 1 - exp(-pt) poi = 1 -pw> pu= 1 -P o, Pooandpu are found by solving the Kolmogorov equations (Eq. 8)... 

We only assume that q is a Markov process, so that its probability distribution obeys the Chapman-Kolmogorov equation... 

Exercise. It has been remarked in 1 that a Markov process with time reversal is again a Markov process. Construct the hierarchy of distribution functions for the reversed Wiener process and verify that its transition probability obeys the Chapman-Kolmogorov equation. 

A careful observation of Eqs. (4.79), (4.80), (4.100) and their respective theoretical basis [4.44, 4.45], allows one to conclude that the probability density distribution that describes the fact that the particle is in position x at t time, when the medium is moving according to one stochastic diffusion process (see relation (4.62) for the analogous discontinuous process), is given by Eq. (4.111). This relation is known as the Eokker-Planck-Kolmogorov equation. 

In Appendix 8A we show that when these conditions are satisfied, the Chapman-Kolmogorov integral equation (8.118) leads to two partial differential equations. The Fokker-Planck equation describes the future evolution of the probability distribution... 

Consider a n-components dynamic system described by an irreducible homogenous Markov process = Xj, t > 0 (initial state /) with finite state space E and the transition rate matrix M. This Markov process is ergodic and a single stationary distribution exists (Ross 1996). Let a row vector jt = (tti, 7T2,. ..) be the vector of steady state probabilities (stationary distribution vector). Chapman-Kolmogorov equations at steady state can be written as ... 

The Kolmogorov-like equations for the absolute probability distribution function can be derived by using assumption (4) ... 

Models of stochastic dynamics (Chapman-Kolmogorov equation for the probability distributions of order parameters) ... 

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... 

The change in concentration of clusters of n molecules may be written as dCn(t)/dt = an-iCn-i(t) — (ccn + Pn)Cn(t) + Pn+iCn+i(t), which has the form of Kolmogorov differential equation for Markov processes in discrete number space and continuous time [21]. and fin are respectively the net probabilities of incorporation or loss of molecules by a cluster per unit time, and these may be defined formally as the aggregation or detachment frequencies times the surface area of the cluster of n molecules. Given the small size of the clusters, and fin are not simple functions of n and in general they are unknown. However, if and fin are not functions of time, then an equilibrium distribution C° of cluster sizes exists, such that dC°/dt = 0 for Cn t) = C°, and the following differential... 

The developed model of has been mathematically described by a linear differential equation system with constant coefficients. It is therefore assumed that the probability of transitions between states is described by exponential distributions, and, consequently, the intensities of transitions between the states are independent of time. The system of forty-four Chapman-Kolmogorov differential equations have been prepared. 

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