The asymptotic behavior of the renormalized zero resolvent of Lévy processes under regular variation conditions

TL;DR

This work characterizes the small-scale asymptotics of the renormalized zero resolvent $h$ for one-dimensional Lévy processes under regular variation assumptions on the Lévy–Khinchin exponents. The authors show that if the real and imaginary parts $\theta$ and $\omega$ of the exponent have a common slowly varying factor and index $\alpha\in\{1,2\}$, then $h(x)$ scales like $|x|^{\alpha-1}$ up to a slowly varying correction, with explicit coefficients determined by $\theta$ and $\omega$. A practical pathway is provided to deduce these regular variation properties from the Lévy measure near 0, yielding concrete conditions and constants for $\theta$ and $\omega$, including the stable-like case. The paper also treats the case with a positive Gaussian component, giving linear small-scale behavior for $h$ and explicit constants involving the imaginary part of $\lambda/\Psi(\lambda)$. An appendix extends the analysis to large arguments, establishing infinity-type asymptotics via regular variation of $\theta$ and $\omega$ near 0 and providing corresponding constants for $h$ at infinity. These results generalize known explicit stable-case formulas to a broad class of Lévy processes and illuminate the penalization framework linked to $h$.

Abstract

As an analogue to the explicit formula in the stable case, the asymptotic behavior at the origin of the renormalized zero resolvent of one-dimensional Lévy processes is studied under certain regular variation conditions on the Lévy-Khinchin exponent and the Lévy measure.

Theorems & Definitions (13)

• Theorem 1.1: Tsukada Tsukada2018
• Theorem 1.2
• Theorem 1.3
• Example 1.4
• Proposition 2.1
• Lemma 2.2
• proof : Proof of Theorem \ref{['thm1.2']}
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm1.3']}
• Theorem 5.1