Lifschitz tail and sausage asymptotics for stable processes in the Poissonian environment on the Sierpinski gasket

TL;DR

This paper investigates the spectral properties and large-time behavior of stable processes on the Sierpinski gasket in a random obstacle environment, revealing Lifschitz tail asymptotics and sausage volume asymptotics.

Contribution

It provides the first analysis of Lifschitz tail asymptotics and sausage volume asymptotics for stable processes on fractals with Poissonian obstacles.

Findings

Lifschitz tail asymptotics for the integrated density of states

Large-time asymptotics for the stable sausage volume

Extension of spectral analysis to fractal environments

Abstract

We obtain the Lifschitz tail asymptotics for the integrated density of states of the subordinate $\alpha-$stable processes on the Sierpi\'nski gasket $\mathcal G,$ evolving among killing Poissonian obstacles. Simultaneously, we derive the large-time asymptotics for the volume of the $\alpha-$stable sausage on the gasket.