The Chapman-Kolmogoroff equation

The simplest random process is completely stochastic so that one may w rite, for example, 

However, here we are concerned with a slightly more complex random process known as a Markov process, characterized by the equation 

The conditional probability serves as some sort of propagator in that it controls the temporal evolution of y t) in the sense of 

The probability distributions satisfy a stationarity condition. In particular, 

Most importantly, Markov processes have a one-step memory . That is, to find y in the interval [y ,7/ -I- dy,, ] at t depends only on the realization y = y i at the immediately preceding time t = t i but is independent of all earlier realizations y = j/m at times t = tm where 1 m n-2. Mathematically speaking this can be cast as 

Let y (t) be a general random process, that is, a process incompletely determined at any given time t. Specific examples are discussed in the context of MC simulations in Section 5.2. The random process can be described by a set of probability distributions P where, for example, P2 J/2 2) dj/idj/S is the probability of finding v/i in the interval [y/i, i f d /i] at time t. = /,j and in the interval [j/2,2/2 + dj/2] at another time t — 2- Thus, the set P forms a hierarchy of probability distributions describing y (f) in increasingly greater detail the larger is n. 

Some properties relevant to the current discussion are listed below  

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As we also point out in Appendix E.1.2, the Chapman-KolmogorofF equation is derived under the assumption of small changes in the random processes represented by y. Hence, the Metropolis algorithm proceeds in two consecutive steps, namely. 

Chapman-Kolmogoroff equation in the theory of stochastic processes. 

It is interesting to notice that a simple formula for the addition of transition probabilities exists only in extremely simple models of stochastic processes. An example is the Chapman-Smoluchowski-Kolmogoroff equation.30... 

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