Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion

Abstract

Motivated by semi-Markov models of anomalous diffusion, we study obliquely reflected diffusions in generalized parabolic domains under independent inverse-subordinator time changes. We focus on infinite-mean subordinators, whose inverse clocks can slow the superdiffusive growth induced by reflection. For domains with cross-sectional radius asymptotic to $a_\infty x^β$, where $-1<β<1$, we strengthen the almost-sure growth laws of [MMW23] to obtain precise second-moment asymptotics when the subordinator's Laplace exponent is regularly varying at zero, and matching bounds more generally. Inverse $α$-stable clocks yield mean-square growth of order $t^{2α/(1+β)}$, giving subdiffusive, diffusive, or superdiffusive regimes. For $β<-1$, explosion persists almost surely, with finite mean explosion time if and only if the subordinator has finite mean. Supporting results establish strong existence, pathwise uniqueness, and a time-change representation, also allowing dependent clocks.