Critical-case second-moment asymptotics at beta equals minus one

Extend the second-moment asymptotic results for the time-changed reflected diffusion to the critical geometric regime beta equals minus one, which is non-explosive but is not covered by the existing results, by establishing the additional moment estimates required for that case.

Background

The paper studies obliquely reflected diffusions in generalized parabolic domains whose cross-sectional radius is asymptotic to a constant times x raised to the power beta, subject to an independent inverse-subordinator time change. For beta between minus one and one, the authors derive second-moment growth estimates, while for beta less than minus one they establish almost-sure explosion and characterize the expected explosion time.

The boundary value beta equals minus one is identified as a critical regime. The authors state that the process is non-explosive there, but their second-moment analysis does not apply. They attribute the difficulty of extending the results to the need for additional moment estimates and leave this extension unresolved.

References

The case $\beta=-1$ is non-explosive, but is not covered by our second-moment results. Extending these results to the critical case requires additional moment estimates and is left for future work.

— Time-changed reflected diffusions: subdiffusion, superdiffusion and explosion  (2610.13302 - Mijatović et al., 9 Oct 2026) in Remark immediately following Proposition \ref{prop:indep subordinator}, Section 1 (Main result)