A local limit theorem for a random walk in an intermittent dynamical environment

Abstract

We study an extended dynamical system on the non-negative real line with piecewise linear non-uniformly expanding local dynamics. With a uniformly distributed initial state, the distribution of successive states coincides with that of a random walk in an inhomogeneous environment. Under suitable conditions on the environment, we establish a central limit theorem and a (non-Gaussian) local limit theorem for the walk. Our approach builds on the work of Leskel\"a and Stenlund (Stochastic Process. Appl. 121(12), 2011), who analyzed a corresponding model with uniformly expanding local dynamics.