The structure of entrance and exit at infinity for time-changed Lévy processes

Studying the behaviour of Markov processes at boundary points of the state space has a long history, dating back all the way to William Feller. Over the past two decades, entrance and exit questions for various discontinuous Markov processes have been explored from different perspectives, often relying on problem-specific knowledge, such as branching or scaling properties. This motivates a key question: How broadly can boundary classifications be established with minimal constraints? We address this by developing sharp criteria for entrance and regular boundary points at infinity in time-changed Lévy processes. Central to our approach is a novel space-time invariance property, which extends classical self-similarity arguments. This property makes use of inherent symmetries to classify boundaries in general settings, and applies to a broad class of time-changed Lévy processes.