On the principal eigenvalue for compound Poisson processes

Abstract

We investigate the explicit expression for the principal eigenvalue $\hat{I}{»}_1^X(D)$ 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 $\hat{I}{»}_1^X(D)$ among all sets with equal Lebesgue measure. Furthermore, we show that this uniqueness fails if the jump density is not strictly decreasing.

There is a critical flaw in the proof and the main theorem, Theorem 2.1, is not true in the stated form