Mittag-Leffler Analysis II: Application to the fractional heat equation

TL;DR

This paper advances Mittag-Leffler analysis by characterizing convergence, approximating delta functions, and applying these methods to construct Green's functions for the fractional heat equation using generalized grey Brownian motion.

Contribution

It extends Mittag-Leffler analysis to distribution spaces and applies it to solve the fractional heat equation via generalized grey Brownian motion.

Findings

Convergent sequences in distribution space characterized.

Donsker's delta approximated by square integrable functions.

Green's function constructed using generalized grey Brownian motion.

Abstract

Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker's delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green's function to the time-fractional heat equation can be constructed using generalized grey Brownian motion (ggBm) by extending the fractional Feynman-Kac formula from Schneider. Moreover we analyse ggBm, show its differentiability in a distributional sense and the existence of corresponding local times.