The paradigm of complex probability and the Brownian motion

Abstract

Andrey N. Kolmogorov's system of axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex probability set which is the sum of the real set with its corresponding real probability, and the imaginary set with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real' laboratory. Whatever the probability distribution of the input random variable in is, the corresponding probability in the whole set is always one, so the outcome of the random experiment in can be predicted totally. The result indicates that chance and luck in is replaced now by total determinism in . This is the consequence of the fact that the probability in is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to the classical theory of Brownian motion and to prove as well the law of large numbers in an original way.