Derive hitting-time asymptotics directly from Varadhan-type heat-kernel asymptotics

Derive the small-time asymptotic behavior of the hitting-time distribution for a nearly Borel target set directly from the Varadhan-type small-time asymptotic of the transition density, rather than through separate sub-Gaussian upper- and lower-bound estimates.

Background

The paper establishes matching small-time logarithmic asymptotics for hitting-time distributions under two-sided sub-Gaussian heat-kernel estimates, with the rate determined by the modified distance from the starting point to the target set. It also notes that the same two-sided estimates imply a Varadhan-type small-time asymptotic for the heat kernel itself.

The authors explicitly identify as unresolved whether the hitting-time asymptotic can be obtained directly from the heat-kernel Varadhan asymptotic, without invoking the intermediate hitting-time estimates developed in the paper. They indicate that this question is intended for future work.

References

A question of particular interest, which remains open, is whether the asymptotic behavior of the hitting time distribution in eq:68 can be derived directly from such a Varadhan-type asymptotic eq:65. We hope to address this problem in future work.

eq:68:

lim⁡t↓0−t1/(β−1)⋅(log⁡Px[σB≤t])C2=d~(x,B)β/(β−1).\lim_{t\downarrow 0}\frac{-t^{1/(\beta-1)}\cdot \left(\log P^x[\sigma_B\leq t]\right)}{C_2}= \widetilde d(x,B)^{\beta/(\beta-1)}.

eq:65:

lim⁡t↓0−t1/(β−1)⋅log⁡p(t,y,z)C2=d(y,z)β/(β−1),y,z∈E.\lim_{t\downarrow 0} \frac{-t^{1/(\beta-1)}\cdot \log p(t,y,z)}{C_2}=d(y,z)^{\beta/(\beta-1)},\quad y,z\in E.

— On hitting time distributions of Markov processes with sub-Gaussian heat kernel bounds  (2608.16170 - Li et al., 17 Aug 2026) in Remark following Corollary 6.3, Section 6