Probability density functions and the dynamics of complex systems associated to some classes of non-archimedean pseudo-differential operators

摘要

In this article, we study certain p-adic master equations which describe the dynamics of a large class of complex systems such as glasses, macromolecules and proteins. These equations are naturally associated to certain non-archimedean pseudo-differential operators whose symbols are connected via Fourier transform with radial probability density functions defined on the p-adic numbers. We show that the fundamental solutions of these equations are probability measures and determine a convolution semigroup on the p-adic numbers. Also, we show that the classical solution of this equations preserves the mass and satisfies the comparison principle. Moreover, we study some strong Markov processes corresponding to radial probability density functions of linear and logarithmic type.