Pseudoprocesses related to space-fractional higher-order heat-type equations

TL;DR

This paper constructs pseudoprocesses related to space-fractional heat equations, demonstrating their convergence and fundamental solutions, extending classical pseudoprocesses to higher real-valued orders.

Contribution

It introduces new pseudoprocesses governed by higher-order space-fractional heat equations involving Riesz and Feller operators, generalizing classical cases.

Findings

Constructed symmetric and asymmetric pseudoprocesses converging to stable subordinated processes.

Derived fundamental solutions matching the laws of these pseudoprocesses.

Extended the class of pseudoprocesses to equations with real order b3 > 2.

Abstract

In this paper we construct pseudo random walks (symmetric and asymmetric) which 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 paper is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order $\gamma>2$. The classical pseudoprocesses are very special cases of those investigated here.