The time-changing of processes by inverse subordinators is widely studied in the literature as models of subdiffusion driven by sporadic trapping. We reveal that increments of such subdiffusion processes may exhibit asymptotic degeneracy depending on the underlying stochastic temporal progression. Various surprising aspects of time-changed processes are discussed in light of asymptotic degeneracy, including diffusive dynamics and optimal inference. We also motivate asymptotic degeneracy as an important criterion for determining appropriateness of such processes for modelling real-world phenomena.

The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, Lévy processes, infinitely divisible processes and fields and Lévy-driven stochastic differential equations. Essential notions and results towards practical implementation are outlined, and summaries of simulation recipes are provided throughout along with numerical illustrations. Some future research directions are highlighted.