Seminario 1 – Semestre 4 de la Red temática de Procesos Estocásticos y sus Aplicaciones.

Speaker: Antonio di Crescenzo (Department of Mathematics, University of Salerno).

Schedule

Abstracts:

Seminar 1: Finite-velocity random motions and reset at the origin: Recent advances on transient and limit behaviors.

Stochastic processes for the description of finite-velocity random motions have been largely studied during the last decades. Usually, they refer to the motion of a particle moving with finite speed on the real line, or on more general domains, with alternations between various possible velocities or directions at random times. The basic model is concerning the so-called (integrated) telegraph process, in which the changes of directions of the two possible velocities are governed by the Poisson process.


In this talk we aim to present some recent results in this research area, involving

1. (i) one-dimensional and two-dimensional finite-velocity random motions, such that the random intertimes between consecutive changes of directions are governed by geometric counting processes,
2. (ii) inclusion of the resetting mechanism to the origin for the one-dimensional process considered in (i),
3. (iii) one-dimensional finite-velocity random motions with instantaneous reset to the origin regulated by Bernoulli trials.

We focus on various features of the considered stochastic processes, including the analysis of the probability laws, the behavior under limit conditions, the mean-square distance between processes, connections with the classical telegraph process.

Seminar 2: Stochastic modeling and applications based on birth-death processes.

Birth-death processes are a flexible tool for stochastic modeling, being a continuous-time analog of random walks. They are largely employed in applications, such as in evolutionary dynamics, neuronal modeling, queueing, reliability and risk theory. The developments in pre-existing techniques for the analysis of birth-death processes have allowed us to achieve new results and new models in this field. Along this direction, the talk will focus on the illustration of recent studies aimed at:

1. (i) discussing growth-evolution models characterized by time-dependent growth rates and their stochastic counterpart described by birth-death processes and diffusive approximations, with special attention to a modified Richards growth model involving a time-dependent perturbation in the growth rate,
2. (ii) the analysis and the application to real data of a two-dimensional time inhomogeneous birth-death process to model the time-evolution of fake news in a population,
3. (iii) the study of the behavior of a multispecies birth-death-immigration process and of a continuous-time multi-type Ehrenfest model, whose diffusion approximations lead to stochastic processes belonging to the class of Pearson diffusions on the spider.

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Red Temática de Procesos Estocásticos y sus Aplicaciones.