A Dirichlet space associated with consistent networks on the ring of p-adic Integers

In this article, we will give a method of reconstruction of a random walk in the ring of p-adic integers. Paying attention to a structural importance in the self similarity, we will perform the construction by means of modified method for constructing canonical stochastic processes on fractals in the Euclidean space. As a result, we will obtain an important subfamily of Albeverio and Karwowski’s stochastic processes with a self-similar randomness.