arXiv subjects

Lorenzo Facciaroni

Publications and source records attributed to Lorenzo Facciaroni.

3 recordsLinked to original sources

arXiv2026

Non-Markovian chains with long-range dependence and their scaling limits

There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some non-Markovian chains, which exhibit long-memory behaviour, due to stochastic dependence among their waiting times. Particular attention is devoted to the so-called para-Markov chains. Their waiting times share the same marginal distributions as those of the above mentioned semi-Markov chains, but they are dependent; their joint distribution is of Schur-constant type and is closely related to complete Bernstein functions and De Finetti's theorems. A second model that we focus on is given by time-changed Markov chains, where the random time is the inverse of an increasing stable process. This generalizes well-known semi-Markov models available in the literature, which typically focus solely on the inverse of the Levy stable subordinator. The above mentioned models are unified by a general theory of time change of Markov chains.

math.PR↗

arXiv2025

Random Flights and Anomalous Diffusion: A Non-Markovian Take on Lorentz Processes

We study Lorentz processes in two different settings. Both cases are characterized by infinite expectation of the free-flight times, contrary to what happens in the classical Gallavotti-Spohn models. Under a suitable Boltzmann-Grad type scaling limit, they converge to non-Markovian random-flight processes with superdiffusive behavior. A further scaling limit yields another non Markovian process, i.e., a superdiffusion obtained by a suitable time-change of Brownian motion. Furthermore, we obtain the governing equations for our random flights and anomalous diffusion, which represent a non-local counterpart for the linear-Boltzmann and diffusion equations arising in the classical theory. It turns out that these equations have the form of fractional kinetic equations in both time and space. To prove these results, we develop a technique based on mixtures of Feller semigroups.

math.PR↗

arXiv2024

Para-Markov chains and related non-local equations

There is a well established theory that links semi-Markov chains having Mittag-Leffler waiting times to time-fractional equations. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times, although marginally Mittag-Leffler, are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes. As a special case of our chains, we study a particular counting process which extends the well-known fractional Poisson process, the last one having independent, Mittag-Leffler waiting times.

math.PR↗