Levy processes and Schroedinger equation

TL;DR

This paper extends the connection between stochastic Levy processes and the Schrödinger equation by introducing a generalized operator, leading to new models including non-stable Levy processes and relativistic quantum equations.

Contribution

It introduces a Levy-Schroedinger equation with a pseudodifferential operator based on Levy process characteristics, generalizing the fractional Schrödinger equation to non-stable Levy processes.

Findings

Recovery of fractional Schrödinger equation for stable Levy processes

Introduction of Levy-Schroedinger equations for non-stable Levy processes

Identification of physically relevant models like relativistic Schrödinger equations

Abstract

We analyze the extension of the well known relation between Brownian motion and Schroedinger equation to the family of Levy processes. We consider a Levy-Schroedinger equation where the usual kinetic energy operator - the Laplacian - is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Levy-Khintchin formula shows then how to write down this operator in an integro--differential form. When the underlying Levy process is stable we recover as a particular case the fractional Schroedinger equation. A few examples are finally given and we find that there are physically relevant models (such as a form of the relativistic Schroedinger equation) that are in the domain of the non-stable, Levy-Schroedinger equations.