text: anybody knows the chapman-kolmogorov equations?   in mathematics, specifically in probability theory, and yet more specifically in the theory of stochastic processes, the chapman-kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process.  suppose that { fi } is an indexed collection of random variables, that is, a stochastic process. let  pi1,...,in(f1 , ... ,fn)  be the joint probability density function of the values of the random variables f1 to fn. then, the chapman-kolmogorov equation is  pi1,...,in-1(f1 , ... ,fn-1) = ∫±∞pi1,...,in(f1 , ... ,fn)dfn  ie a straightforward marginalization over the nuisance variable.  (note that we have not yet assumed anything about the temporal (or any other) ordering of the random variables -- the above equation applies equally to the marginalization of any of them).