Incompatible Stochastic Processes and Complex Probabilities

[ The definition of conditional probabilities is based upon the existence of a joint A probability. However, a reconstruction of the joint probability from given conditional probabilities imposes certain constraints upon the latter, so that if several condit.iomd probabilities are chosen arbitrarily, the corresponding joint probability may not exist, Such an incompleteness in conditional probabilities can be eliminated by introducing complex probabilities. Physical meaning of the new mathematical formalism, as well as its relation to quantum probabilities, is discussed. One of the oldest and still processes is to reconstruct a unsolved problems in the field of rnultivariate stochastic joint probability from several correlated conditional probabilities. This problem has been discussed in [1] [3]. Its origin is in the fact that classical probability theory defines conditional probabilities based upon the existence of a joint probability. At the same time, one can observe correlated stochastic processes which are represented by conditional probabilities. And then the inverse problem of reconstructing the underlying joint probability arises. As an illustration to this point, consider two coupled diffusion equations: a2p, (x, ,X2) ~P2 (xl JX2 ) = ~ (x, ) ~2d~l ‘Z f3p1(x,,x*) =D,(X2) ~x2 ,~[ 2 &’ ) (2 1 2 with the initial conditions: p,(x,>t’lx~>t’) = 6(X, x:), i = 1 , 2 The solution for t > t’ reads: (1)