Selfdecomposability and selfsimilarity: a concise primer

We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Levy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.Comment: 24 pages, 3 figures; corrected misprint in the title; redactional modifications required by the referee; added references from [16] to [28];. Accepted and in press on Physica

Thou shalt not say "at random" in vain: Bertrand's paradox exposed

We review the well known Bertrand paradoxes, and we first maintain that they do not point to any probabilistic inconsistency, but rather to the risks incurred with a careless use of the locution "at random". We claim then that these paradoxes spring up also in the discussion of the celebrated Buffon's needle problem, and that they are essentially related to the definition of (geometrical) probabilities on "uncountably" infinite sets. A few empirical remarks are finally added to underline the difference between "passive" and "active" randomness, and the prospects of any experimental decisionComment: 17 pages, 4 figures. Added: Appendix A; References 7, 8, 10; Modified: Abstract; Section 4; a few sentences elsewher

Phenomenology from relativistic Levy-Schroedinger equations: Application to neutrinos

In continuation of a previous paper a close connection between Feynman propagators and a particular L\'evy stochastic process is established. The approach can be easily applied to the Standard Model SU_C(3)xSU_L(2)xU(1) providing qualitative interesting results. Quantitative results, compatible with experimental data, are obtained in the case of neutrinos

Controlled quantum evolutions and transitions

We study the nonstationary solutions of Fokker-Planck equations associated to either stationary or nonstationary quantum states. In particular we discuss the stationary states of quantum systems with singular velocity fields. We introduce a technique that allows to realize arbitrary evolutions ruled by these equations, to account for controlled quantum transitions. The method is illustrated by presenting the detailed treatment of the transition probabilities and of the controlling time-dependent potentials associated to the transitions between the stationary, the coherent, and the squeezed states of the harmonic oscillator. Possible extensions to anharmonic systems and mixed states are briefly discussed and assessed.Comment: 24 pages, 4 figure