Università degli Studi di Salerno Dipartimento di Matematica

Si rende noto che nel periodo 25 giugno - 12 luglio 2019 si terrà il corso di dottorato

"From Markov chains to semi-Markov processes: The consequences of time change".

Le lezioni saranno tenute dal Prof. Enrico Scalas, Professor of Statistics & Probability, University of Sussex, UK

(pagina web: http://www.sussex.ac.uk/profiles/330303)

e si terranno nella sala del consiglio del Dipartimento di Matematica secondo il seguente calendario:

- lezione 1 (2 ore): martedì 25 giugno, ore 11:00-13:00;

- lezione 2 (3 ore): venerdì 28 giugno, ore 10:00-13:00;

- lezione 3 (3 ore): martedì 2 luglio, ore 10:00-13:00;

- lezione 4 (3 ore): giovedì 4 luglio, ore 10:00-13:00;

- lezione 5 (2 ore): martedì 9 luglio, ore 11:00-13:00;

- lezione 6 (3 ore): venerdì 12 luglio, ore 10:00-13:00.

Gli interessati sono invitati a partecipare.

I contenuti del corso sono riportati qui di seguito.

Research context

What happens if the discrete time at which innovations occur for a Markov chain is replaced by a counting process? If the counting process is the Poisson process, one still gets a Markov process. On the contrary, if the counting process is of renewal type, the resulting process is no longer Markovian: It becomes a semi-Markov process. Semi-Markov processes have infinite memory and are naturally related to non-local operators in time such as the so-called Caputo derivative that appears when the expected values of inter-event times diverge. For this reason, there has been a surge of interest on these processes in recent years.

Syllabus:

1. From Markov chains to semi-Markov processes: General framework (2 hours)

1.1 From the Poisson process to the fractional Poisson process

1.2 Time change and its consequences

2. Example 1: Continuous-time random walks (CTRWs) and applications (8 hours)

2.1 The diffusion equation and the normal compound Poisson process

2.2 The space-time fractional diffusion equation and CTRWs

2.3 Some applications

3. Example 2: Semi-Markov graph dynamics (6 hours)

Pre-requisites

The course will be accessible to first year PhD students (Corso di Dottorato in Matematica, Fisica e Applicazioni). In fact, the main pre-requisites are:

1. An introductory course in probability and statistics.

2. Fourier and Laplace transforms.

Learning objectives

After following this course, the students should be able:

1. To read and understand the recent literature on semi-Markov process

2. To perform calculations on continuous-time random walks and other semi-Markov processes using elementary probabilistic methods

Essential references:

- Scalas, Enrico (2017) Continuous-time statistics and generalized relaxation equations. European Physical Journal B: Condensed Matter and Complex Systems, 90 (11). p. 209. ISSN 1434-6028.

- Baleanu, Dumitru, Diethelm, Kai, Scalas, Enrico and Trujillo, Juan J (2016) Fractional calculus: models and numerical methods (2nd edition). Series on complexity, nonlinearity and chaos, 5. World Scientific, Singapore. ISBN 9789813140035.

- Georgiou, Nicos, Kiss, Istvan Z and Scalas, Enrico (2015) Solvable non-Markovian dynamic network. Physical Review E, 92 (4). 042801. ISSN 1539-3755.