2008/05/30 by Nicola Cufaro Petroni, M. Pusterla, Modesto Pusterla · 53 citations
Mathematics · Physics and Astronomy · #Applied mathematics #Fractional Differential Equations Solutions #Mathematical physics #Mathematics #Physics #Stochastic processes and statistical mechanics #advanced mathematical theories #cond-mat.stat-mech #math.PR #quant-ph
paper · pdf · doi:10.1016/j.physa.2008.11.035
published in Physica A Statistical Mechanics and its Applications 388(6), 824-836 (Elsevier BV) · 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
arxiv created 2008/05/30 · openalex publication_date 2008/12/04 · arxiv updated 2014/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.