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

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).