Quenched asymptotics for symmetric L\'evy processes interacting with Poissonian fields

TL;DR

This paper derives explicit long-time asymptotic behaviors for symmetric Lévy processes interacting with Poissonian fields, revealing detailed insights into the solutions of related nonlocal parabolic Anderson problems.

Contribution

It provides the first explicit quenched asymptotics for a broad class of symmetric Lévy processes in Poissonian potentials, including critical cases.

Findings

Explicit quenched asymptotics for Lévy processes with specific Lévy measures.

Asymptotic behavior characterized for potentials with shape function involving power-law decay.

Discussion of quenched asymptotics in critical cases like β=2.

Abstract

We establish explicit quenched asymptotics for pure-jump symmetric L\'evy processes in general Poissonian potentials, which is closely related to large time asymptotic behavior of solutions to the nonlocal parabolic Anderson problem with Poissonian interaction. In particular, when the density function with respect to the Lebesgue measure of the associated L\'evy measure is given by $$\rho(z)= \frac{1}{|z|^{d+\alpha}}\I_{\{|z|\le 1\}}+ e^{-c|z|^\theta}\I_{\{|z|> 1\}}$$ for some $\alpha\in (0,2)$, $\theta\in (0,\infty]$ and $c>0$, exact quenched asymptotics is derived for potentials with the shape function given by $\varphi(x)=1\wedge |x|^{-d-\beta}$ for $\beta\in (0,\infty]$ with $\beta\neq 2$. We also discuss quenched asymptotics in the critical case (e.g.,\, $\beta=2$ in the example mentioned above).