On Markovian behaviour of $p$-adic random dynamical systems

Abstract

We study Markovian and non-Markovian behaviour of stochastic processes generated by $p$-adic random dynamical systems. Given a family of $p$-adic monomial random mappings generating a random dynamical system. Under which conditions do the orbits under such a random dynamical system form Markov chains? It is necessary that the mappings are Markov dependent. We show, however, that this is in general not sufficient. In fact, in many cases we have to require that the mappings are independent. Moreover we investigate some geometric and algebraic properties for $p-$adic monomial mappings as well as for the $p-$adic power function which are essential to the formation of attractors. $p$-adic random dynamical systems can be useful in so called $p$-adic quantum phytsics as well as in some cognitive models.