Pseudoprocesses governed by higher-order fractional differential equations

TL;DR

This paper investigates higher-order fractional heat equations, revealing that their solutions can be interpreted as pseudoprocesses with a random time argument, and provides explicit distribution and moment formulas.

Contribution

It introduces a novel interpretation of solutions to higher-order fractional heat equations as pseudoprocesses with random time, including explicit distribution and moment characterizations.

Findings

Solution described as pseudoprocess with random time T

Explicit distribution of T provided

Analytic expressions for moments of the solution

Abstract

We study here a heat-type differential equation of order n greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess (coinciding with the one governed by the standard, non-fractional, equation) with a time argument T which is itself random. The distribution of T is presented together with some features of the solution (such as analytic expressions for its moments).