Discrete Time Processes Embedded in Continuous Time Systems

Abstract

In this chapter, our goal is to introduce a way in which discrete time processes may be embedded in continuous time systems without altering the phase space. To do this, we adopt a strictly probabilistic point of view, not embedding the deterministic system S: X → X in a continuous time process, but rather embedding its Frobenius-Perron operator P:L ¹ (X) →L ¹ (X) that acts on L ¹ functions. The result of this embedding is an abstract form of the Boltzmann equation. This chapter requires some elementary definitions from probability theory and a knowledge of Poisson processes, which are introduced following the preliminary remarks of the next section.