Levy processes and Schroedinger equation

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.

10 pages; changed the TeX documentclass; added references [21] and [22] and comments about them; changed definitions (11) and (12); added acknowledgments; small changes scattered in the text

References in corpus (1)

• On some results of Cufaro Petroni about Student t-processes

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