Infinite Invariant Measure in a Heterogeneous Accumulation Process (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

Abstract

We provide an exact form of the infinite invariant density in a semi-Markov process. In the semi-Markov process, the state almost surely converges to zero in the long-time limit, i.e., accumulation to the origin, and the infinite invariant density characterizes the accumulation process. We demonstrate two distributional behaviors for time averages of the absolute value of the state, where the shape of the distribution depends on whether the observable is integrable with respect to the infinite invariant density. Therefore, the infinite invariant density plays an important role in characterizing the non-stationary accumulation process.