Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise

TL;DR

This paper introduces Gamma-grey noise, a new non-Gaussian measure in infinite-dimensional spaces, and explores its properties, related processes, and applications to modeling anomalous diffusions using fractional calculus.

Contribution

It constructs Gamma-grey noise using the incomplete-gamma function, establishes the existence of an Appell system, and defines associated generalized processes and Brownian motions.

Findings

Existence of Appell system for Gamma-grey noise

Characterization of Gamma-grey Brownian motion

Derivation of integro-differential equations for transition densities

Abstract

In the context of non-Gaussian analysis, Schneider [27] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [25], [6], [7], [8]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure Gamma-grey noise and we prove, for it, the existence of Appell system. The related generalized processes, in the infinite dimensional setting, are also defined and, through the use of the Riemann-Liouville fractional operators, the (possibly tempered) Gamma-grey Brownian motion is consequently introduced. A number of different characterizations of these processes are also provided, together with the integro-differential equation satisfied by their transition densities. They allow to model anomalous diffusions, mimicking the procedures of classical stochastic calculus.