Determine the slower-rate asymptotics without the Pascal-principle condition

Establish matching upper bounds and determine whether the annealed survival probability for the inhomogeneous mobile-trap random walk, when Assumption (A2) fails but Assumption (A3) holds—as in the profile \(\nu_y=1/(1+|y|^\beta)\) in dimension one with \(0<\beta<1\)—decays at the slower rate \(\exp\{-C t^{(1-\beta)/2}\}\).

Background

The paper proves sharp annealed-survival asymptotics in dimensions one and two under bounded initial intensities and the condition that the mean trap intensity is minimized at the origin (Assumption (A2)). It also considers one-dimensional profiles for which (A2) fails but the weaker regular-variation condition (A3) holds, including $\nu_y=1/(1+|y|^\beta)$ for $0<\beta<1$. For this example, the paper establishes only a lower bound with decay scale $t^{(1-\beta)/2}$, which is slower than the $t^{1/2}$ scale obtained under (A2).

The unresolved issue is whether this lower-bound scale is the true asymptotic decay rate. The authors explain that they cannot obtain matching upper bounds because their current Pascal-principle proof requires (A2), and they further suggest that weakening (A2) may lead to qualitatively different asymptotic behavior.