Generalized random processes related to Hadamard operators and Le Roy measures

TL;DR

This paper introduces generalized random processes using Hadamard-type fractional operators, extending stochastic models like fractional Brownian motion, and explores their properties, long-term behavior, and connections to Le Roy measures and heat equations.

Contribution

It constructs Hadamard fractional Brownian motion and extends it to grey-noise spaces based on Le Roy measures, providing new insights into their distributional properties and associated PDEs.

Findings

Hadamard fractional Brownian motion exhibits standard diffusion over finite times.

Extension to Le Roy measure yields processes satisfying heat equations with fractional derivatives.

Constructed Ornstein-Uhlenbeck process and analyzed its distribution.

Abstract

The definition of generalized random processes in Gel'fand sense allows to extend well-known stochastic models, such as the fractional Brownian motion, and study the related fractional pde's, as well as stochastic differential equations in distributional sense. By analogy with the construction (in the infinite-dimensional white-noise space) of the latter, we introduce two processes defined by means of Hadamard-type fractional operators. When used to replace the time derivative in the governing p.d.e.'s, the Hadamard-type derivatives are usually associated with ultra-slow diffusions. On the other hand, in our construction, they directly determine the memory properties of the so-called Hadamard fractional Brownian motion (H-fBm) and its long-time behaviour. Still, for any finite time horizon, the H-fBm displays a standard diffusing feature. We then extend the definition of the H-fBm from the white noise space to an infinite dimensional grey-noise space built on the Le Roy measure, so that our model represents an alternative to the generalized grey Brownian motion. In this case, we prove that the one-dimensional distribution of the process satisfies a heat equation with non-constant coefficients and fractional Hadamard time-derivative. Finally, once proved the existence of the distributional derivative of the above defined processes and derived an integral formula for it, we construct an Ornstein-Uhlenbeck type process and evaluate its distribution.