The paradigm of complex probability and the Brownian motion

What is it about?

The 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 C which is the sum of the real set R with its corresponding real probability, and the imaginary set M 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 R is, the corresponding probability in the whole set C is always one, so the outcome of the random experiment in C can be predicted totally. The result indicates that chance and luck in R is replaced now by total determinism in C. This is the consequence of the fact that the probability in C 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.

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Why is it important?

All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, the probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new dimensions to the event in R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities C. It is of great importance that the stochastic system, like the problem considered here, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature for example in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities which will make the event in C = R +M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and prognostic is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This is what I called ‘The Complex Probability Paradigm (CPP)’, which was initiated and elaborated in my previous papers.

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