TOWARD THEORY OF P-ADIC VALUED PROBABILITIES

We present a short review on generalization of probability theory in that probabilities take values in the fields of p-adic numbers, Qp. Such probabilities were introduced to serve p-adic theoretical physics. In some quantum physical models a wave function (which is a complex probability amplitude in ordinary QM) takes vales in Qp (for some prime number p) or their quadratic extensions. Such a wave function can be interpreted probabilistically in the framework of p-adic probability theory. This theory was developed by using both the frequency approach (by generalizing von Mises) and the measure-theoretic approach (by generalizing Kolmogorov). In particular, some limit theorems were obtained. However, theory of limit theorems for p-adic valued probabilities is far from being completed. Another interesting domain of research is corresponding theory of complexity. We obtained some preliminary results in this direction. However, it is again far from to be completed. Recently p-adic models of classical statistical mechanics were considered and some preliminary results about invariant p-adic valued measures for dynamical systems were obtained.