The Faculty of Natural and Applied Sciences, Department of Mathematics and Statistics at Notre Dame University-Shouf Campus, organized a lecture under the title of “The Complex Probability Paradigm and the Brownian Motion.” The lecture, presented by Dr. Abdo Abou Jaoudé was held on Friday, March 17, 2017, in the conference room.
Dr. Abdo Abou Jaoudé explained the five basic axioms of Andrey Nikolaevich Kolmogorov which can be extended to encompass the imaginary set of numbers by adding to the original five axioms an additional three axioms. Hence, any experiment can 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.
Dr. Abou Jaoudé added that the objective is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the "real" laboratory. Therefore, 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. Knowing that 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 was applied to the classical theory of the Brownian motion and proved as well the law of large numbers in an original way.
This result was elaborated throughout the lecture using the numerical example of the diffusion of oxygen gas in air gas. In addition, the lecture further developed the new field of the “Complex Probability Paradigm” created by Dr. Abou Jaoudé and that considers all the random variables in the complex set C. Thus, the application to the Brownian motion proves that this complex extension is both successful and fruitful.