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A note on geometric α-stable processes and the existence of ground states for associated Schrödinger operators

Kaneharu Tsuchida

Abstract

In this paper, we establish the existence of transition density for geometric $α$-stable processes by using the property of self-decomposability--a fundamental concept in the theory of Lévy 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 Lévy measure. As an application, we prove the existence of ground states for Schrödinger operators associated with recurrent geometric stable processes.

A note on geometric α-stable processes and the existence of ground states for associated Schrödinger operators

Abstract

In this paper, we establish the existence of transition density for geometric -stable processes by using the property of self-decomposability--a fundamental concept in the theory of Lévy processes. In contrast to traditional and analytic methods that often rely on the -integrability of the characteristic function, our approach is purely probabilistic and focuses on the structural regularity of the Lévy measure. As an application, we prove the existence of ground states for Schrödinger operators associated with recurrent geometric stable processes.
Paper Structure (4 sections, 9 theorems, 29 equations)

This paper contains 4 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of the transition density
  4. Application to the existence of ground states

Key Result

Theorem 2.1

Let $J$ be the Lévy measure associated with the geometric $\alpha$-stable process ${\mathbf{M}}$. For every $\alpha \in (0,2]$, there exists a density function $j(x)$ of $J$ with respect to the Lebesgue measure. Moreover, the following holds:

Theorems & Definitions (16)

  • Theorem 2.1: SikicSongVondracek2006
  • Definition 3.1
  • Lemma 3.2: Sato1999Levy
  • Lemma 3.3: Sato1999Levy
  • Theorem 3.4
  • proof
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3: Takeda2019
  • Corollary 4.4
  • ...and 6 more