A note on geometric {\alpha}-stable processes and the existence of ground states for associated Schr\"odinger operators

TL;DR

This paper proves the existence of transition densities for geometric alpha-stable processes using probabilistic methods based on self-decomposability, and demonstrates the existence of ground states for related Schrödinger operators.

Contribution

It introduces a purely probabilistic approach to establish transition densities and ground states for geometric stable processes, diverging from traditional analytic methods.

Findings

Transition densities exist for geometric alpha-stable processes.

Ground states exist for Schrödinger operators with recurrent geometric stable processes.

Probabilistic methods can replace analytic techniques in this context.

Abstract

In this paper, we establish the existence of transition density for geometric $\alpha$-stable processes by using the property of self-decomposability--a fundamental concept in the theory of L\'evy processes. In contrast to traditional and analytic methods that often rely on the $L^{1}$-integrability of the characteristic function, our approach is purely probabilistic and focuses on the structural regularity of the L\'evy measure. As an application, we prove the existence of ground states for Schr\"odinger operators associated with recurrent geometric stable processes.