Removal of the logarithmic perturbation factor

Determine whether the logarithmic factor in the function \(\vartheta(z)=z\log(1+z^{-1})\) can be removed from the perturbation bounds for Feynman–Kac models under the conditions of Section 3, so that the global error is bounded proportionally to the largest local perturbation.

Background

The paper establishes global perturbation estimates of order $\vartheta(\Delta)$, where $\Delta$ is the largest local error. The logarithmic factor arises because the exponential forgetting estimate does not scale with the initial discrepancy, whereas the local Lipschitz estimate scales with that discrepancy but does not decay over time. Summing the resulting minimum of these two bounds produces the logarithmic term.

The authors note that stronger uniform minorization assumptions can yield exponentially decaying Lipschitz constants and remove this factor, but those assumptions are unavailable under the paper’s more general conditions, which allow non-compact state spaces and unstable Markov transitions. The question is therefore whether the logarithmic loss is intrinsic under the stated conditions.