Chapter 3 - Recurrence and transience

Abstract

This chapter discusses recurrence and transience. The dynamic random walk can be termed as recurrent if each and every point is recurrent or transient. Assumptions can be made that the dynamic random walk is recurrent in dimension d. The proofs rest on the central limit theorem and on the local limit theorem. The dynamical system can be assumed ergodic. The first assertion can be easily proved by using the strong law of large numbers of integrals for the dynamic random walk. The dynamic random walk implies the transience. In higher dimension, the transience can be followed by using the strong of large numbers of integrals for the dynamic random walks. The local limit theorem implies that there exists a strictly positive constant C. The denominator is equivalent to log (n). The recurrence of the two-dimensional dynamic random walk can take place in the local limit theorem. When all the integrals are equal to each other, that case becomes more interesting.