Mittag-Leffler Analysis: Construction and Applications

Motivated by the results of infinite dimensional Gaussian analysis and especially white noise analysis, we construct a Mittag-Leffler analysis. This is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. Our results indicate that the Wick ordered polynomials, which play a key role in Gaussian analysis, cannot be generalized to this non-Gaussian case. We provide evidence that a system of biorthogonal polynomials, called generalized Appell system, is applicable to the Mittag-Leffler measures, instead of using Wick ordered polynomials. With the help of an Appell system, we introduce a test function and a distribution space. Furthermore we give characterizations of the distribution space and we characterize the weakly integrable functions and the convergent sequences within the distribution space. We construct Donsker's delta in a non-Gaussian setting as an application. In the second part, we develop a grey noise analysis. This is a special application of the Mittag-Leffler analysis. In this framework, we introduce generalized grey Brownian motion and prove differentiability in a distributional sense and the existence of generalized grey Brownian motion local times. Grey noise analysis is then applied to the time-fractional heat equation and the time-fractional Schrödinger equation. We prove a generalization of the fractional Feynman-Kac formula for distributional initial values. In this way, we find a Green's function for the time-fractional heat equation which coincides with the solutions given in the literature. The results presented in this talk are published in the references below.

References: 

M. Grothaus, F. Jahnert, F. Riemann and J. L. da Silva: Mittag-Leffler analysis I: Construction and characterization. Journal of Functional Analysis, 268(7), 1876--1903, 2015.

M. Grothaus and F. Jahnert: Mittag-Leffler analysis II: Application to the fractional heat equation. Journal of Functional Analysis, 270(7), 2732--2768, 2016.

The speaker is member of a Research in Pairs program at CIRM.