Non-Archimedean probability: Frequency and axiomatics theories

Abstract

We propose a new theory of probability based on the general principle of the statistical stabilization of relative frequencies. According to this principle it is possible to consider the statistical stabilization not only with respect to the standard real topology on the field of rational numbersQ but also with respect to an arbitrary topology onQ. The case ofp-adic (and more general non-Archimedean) topologies is the most important. Our frequency theory of Probability is a fruitful extension of the frequency theory of R. von Mises[18]. It's well known that the axiomatic theory of Kolmogorov uses the frequency theory as one of the foundations. And a new general frequency theory can be considered as the base for the general axiomatic theory of probability (Kolmogorov's theory is a particular case of this theory which corresponds to the real topology of the statistical stabilization onQ). The situation in the theory of probability becomes similar to that in modern geometry. The Kolmogorov axiomatics (as the Euclidean) is only one of the possibilities, and we have generated a great number of different non-Kolmogorov theories of probability. The applications top-adic quantum mechanics and field theory are considered.