Chapman-Kolmogorov model

Using the Chapman-Kolmogorov equation (1) for Markov analysis, it is possible to analyse this model in order to determine the probabilities, P(t), of each failure mode. 

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. 

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

Let us suppose that the posterior pdf p Xf. I at time A — 1 is available and let us assume that Ihe noise sequences and are white, with known probability density functions and mutually independent. Moreover, let us assume thatdie initial state vector hasaknown pdf p xq ) and is also independent of noise sequences. The prediction stage evolves using the system model to obtain the prediction density of the state attime k viathe Chapman-Kolmogorov equation ... 

---