Geometric Stable processes and related fractional differential equations

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha}^{\beta}=\left\{\mathcal{G}_{\alpha}^{\beta}(t);t\geq 0\right\}$, with stability \ index $$ and asymmetry parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_{\alpha}^{\beta}.$ For some particular values of $$ and $\beta ,$ we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.Comment: 12 page

From cosmic ray physics to cosmic ray astronomy: Bruno Rossi and the opening of new windows on the universe

Bruno Rossi is considered one of the fathers of modern physics, being also a pioneer in virtually every aspect of what is today called high-energy astrophysics. At the beginning of 1930s he was the pioneer of cosmic ray research in Italy, and, as one of the leading actors in the study of the nature and behavior of the cosmic radiation, he witnessed the birth of particle physics and was one of the main investigators in this fields for many years. While cosmic ray physics moved more and more towards astrophysics, Rossi continued to be one of the inspirers of this line of research. When outer space became a reality, he did not hesitate to leap into this new scientific dimension. Rossi's intuition on the importance of exploiting new technological windows to look at the universe with new eyes, is a fundamental key to understand the profound unity which guided his scientific research path up to its culminating moments at the beginning of 1960s, when his group at MIT performed the first in situ measurements of the density, speed and direction of the solar wind at the boundary of Earth's magnetosphere, and when he promoted the search for extra-solar sources of X rays. A visionary idea which eventually led to the breakthrough experiment which discovered Scorpius X-1 in 1962, and inaugurated X-ray astronomy.Comment: This work was presented at the conference "100 Years Cosmic Ray Physics - Anniversary of the V.F. Hess Discovery", 6-8 August, Bad Saarow/Pieskow, Germany, where Hess landed on August 7, 1912, after discovery of the "H\"ohenstrahlung". To be published in the Astroparticle Journa