On the principal eigenvalue for compound Poisson processes

TL;DR

This paper studies the principal eigenvalue of compound Poisson processes on bounded domains, showing that for radially symmetric decreasing jump densities, balls uniquely minimize the eigenvalue, but this property does not hold otherwise.

Contribution

It provides explicit formulas for the principal eigenvalue and characterizes the geometric minimizers for specific jump densities of compound Poisson processes.

Findings

Balls are the unique minimizers for radially symmetric decreasing jump densities.

The uniqueness of minimizers fails if the jump density is not strictly decreasing.

Explicit spectral heat content expressions are derived for the processes.

Abstract

We investigate the explicit expression for the principal eigenvalue $\lambda_{1}^{X}(D)$ for a large class of compound Poisson processes $X$ on a bounded open set $D$ by examining its spectral heat content. When the jump density of the compound Poisson process is radially symmetric and strictly decreasing, we demonstrate that balls are the unique minimizers for $\lambda_{1}^{X}(D)$ among all sets with equal Lebesgue measure. Furthermore, we show that this uniqueness fails if the jump density is not strictly decreasing.