Generalized random processes related to Hadamard operators and Le Roy measures
Luisa Beghin, Lorenzo Cristofaro, Federico Polito
TL;DR
This work develops a framework for generalized random processes via Hadamard-type fractional calculus in Gel'fand triples, introducing Hadamard fractional Brownian motion (H-fBm) and its Le Roy extension (LHm). It shows that one-dimensional marginals remain Gaussian with variance $t$ while memory properties and long-time behaviour depend on the Hadamard parameter $\alpha$, and extends to Le Roy–Hadamard processes parameterized by $\beta$ through the Le Roy function, yielding a fractional heat equation with Hadamard time derivative. The authors establish distributional derivatives, integral representations, and stochastic-analysis tools in the Le Roy grey-noise setting, including a rigorous $S_{\nu_{\beta}}$-calculus and test/distribution spaces. They further construct an LHm-driven Ornstein-Uhlenbeck process and derive its distribution, illustrating how heterogeneous, nonlocal, memory-laden dynamics arise from Hadamard and Le Roy mechanisms. Overall, the paper connects Hadamard fractional calculus with infinite-dimensional stochastic processes and nonlocal diffusion equations, offering a modeling framework for systems with ultra-slow diffusion features and complex memory.
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.