Fractional generalization of Kac integral
type: Item
instance of: scholarly article
title: Fractional generalization of Kac integral
zbMATH Open document ID: 1133.26004
author: Vasily E. Tarasov
author: George M. Zaslavsky
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2007-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0704.1771
review text: \textit{M. Kac}'s integral [Bull. Am. Math. Soc. 54, 64 (1948); Trans. Am. Math. Soc. 65, 1--13 (1949; Zbl 0032.03501)] appeared as a path-wise presentation of Brownian motion, and with \textit{R.P. Feynman}'s approach [Rev. Mod. Phys. 20, 367--387(1948)], became a powerful tool to study different processes described by the wave-type or diffusion-type equations. Recently the Lévy measure was applied to derive a fractional generalization of the Schrödinger equation, using the Feynman-type approach and expressing the Lévy measure through the Fox \(H\)-function. In this paper the authors derive the fractional generalization of the diffusion equation from the path integral over the Lévy measure using the integral equation approach of Kac. A function introduced in the analysis is expressed in terms of Fox \(H\)-function.
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 35K99
Mathematics Subject Classification ID: 33C60
zbMATH DE Number: 5209170
zbMATH Keywords: Kac path integral
zbMATH Keywords: Lévy distribution
zbMATH Keywords: fractional derivative
zbMATH Keywords: fractional Ginzburg-Landau equation
reviewed by: S. L. Kalla
MaRDI profile type: Publication
OpenAlex ID: W1972859147
arXiv ID: 0704.1771
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DOI: 10.1016/J.CNSNS.2007.04.020
description: scientific article