Pseudoprocesses Related to Space-Fractional Higher-Order Heat-Type Equations

Abstract

In this article, we construct pseudo random walks (symmetric and asymmetric) that converge in law to compositions of pseudoprocesses stopped at stable subordinators. We find the higher-order space-fractional heat-type equations whose fundamental solutions coincide with the law of the limiting pseudoprocesses. The fractional equations involve either Riesz operators or their Feller asymmetric counterparts. The main result of this article is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order γ > 2. The classical pseudoprocesses are very special cases of those investigated here.

Keywords:

• Weyl derivatives
• Riesz derivatives
• Feller fractional operators
• Pseudoprocesses
• Stable processes
• Subordinators
• Continuous-time random walks

2000 Mathematics Subject Classification:

• 60K99
• 35Q99

Acknowledgments

The authors have benefited from fruitful discussions on the topics of this article with Dr. Mirko D’Ovidio. Thanks are due to the referee whose remarks and suggestions have certainly improved the article.