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Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion

  • Aleksandar Mijatović
  • Francesco Virgili
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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∞ xβ, 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 t2α/(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.

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Cite

BibTeXRIS
BibTeX
@misc{mijatovic2026timechanged,
  title = {{Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion}},
  author = {Aleksandar Mijatović and Francesco Virgili},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13302},
  eprint = {2610.13302},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13302},
}
RIS
TY  - UNPB
TI  - Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion
AU  - Aleksandar Mijatović
AU  - Francesco Virgili
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13302
UR  - https://arxiv.org/abs/2610.13302
AB  - 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_∞ 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.
ER  -