Brownian fluctuation limit in the excluded logarithmic case

Establish the Brownian fluctuation limit for the Lévy-driven slow-fast system with slow drift f(y)=y^q and fast drift g(y)=-|y|^p\operatorname{sgn}(y) in the excluded case r=q+1-p=0, using the logarithmic corrector \Phi_0(y)=-\operatorname{sgn}(y)\log|y| and determining the corresponding quantitative estimates and normalization.

Background

The main theorem excludes the parameter value p=q+1, equivalently r=q+1-p=0, because the corrector \Phi(y)=-(1/r)|y|r\operatorname{sgn}(y) is not defined there. The authors indicate that the Brownian-limit result in part (ii), which applies when r<\alpha/2, is expected to remain valid at r=0 after replacing the corrector by the logarithmic function \Phi_0(y)=-\operatorname{sgn}(y)\log|y|.

A proof at r=0 would require rewriting quantitative estimates used in the existing argument, specifically the Hölder estimate and the technical estimate identified as Lemma \ref{lemma-Holder} and Lemma \ref{lemma-technique}. Thus, the unresolved task is to prove the claimed Brownian fluctuation limit in this excluded parameter regime and supply those missing estimates.

References

It is expected that Theorem \ref{main} (ii) is valid in $r=0$ case, although we exclude this case in our paper. Defining $\Phi_0(y):=-\operatorname{sgn}(y)\log|y|,$ and replacing $\Phi$ by $\Phi_0,$ a similar proof with that in Section \ref{subsection-r<alpha/2} is expected to work. However, several quantitative estimates have to be re-written, such as Lemma \ref{lemma-Holder} and Lemma \ref{lemma-technique}. This is left to subsequent work.

— Stable and Gaussian Fluctuation Limit of a Lévy-Driven Slow-Fast System with Polynomial Dissipation  (2609.09906 - Zhao et al., 9 Sep 2026) in Remark following Theorem 1, immediately before the organization of the paper