French
Summary of the RESYST project
The RESYST project is interested in reflected stochastic processes involved in several systems of queueing networks. These stochastic models have been developed due to their numerous applications in operations research, risk theory, telecommunication, data sciences and also in population biology.
The issues raised by such stochastic systems are intertwined with many fields of mathematics (probability, complex analysis, combinatorics, differential Galois theory).
A typical example of such a continuous process is obliquely reflected Brownian motion in an orthant. Their discrete analogues, random walks in cones, are also very famous processes. There are a wide variety of interesting questions regarding these reflected processes which can be either recurrent or transient. Among these questions there are the study of invariant measures, Green’s functions, time to reach an edge or the vertex, algebraic nature of the generating function, harmonic functions and Martin boundary associated to the process.
A now standard method to study these problems is to establish kernel functional equations involving generating functions. The project intends to decompartmentalize certain approaches by mixing three different techniques used to solve these functional equations: Boundary value problems, Tutte’s invariant approach and q-difference equations.
The project’s objectives are to study:
1. Persistence, extinction and quasi-stationary distribution in a cone.
2. Ruin and escape for transient process in an orthant.
3. Monte Carlo simulation for Brownian motion in cones.
4. Additive and multiplicative decoupling for Tutte’s invariant approach.
5. Transition kernel and space-time functional equation.
JCJC ANR Project
Acronym: RESYST (Reflected Stochastic Systems)
Period: 2023/2025
Budget: 178.300 euros
Project number: ANR-22-CE40-0002
Scientific coordinator
Sandro Franceschi (Associate professor, Télécom SudParis, Institut Polytechnique de Paris)
PhD Student
Jules Flin (thesis grant funded by ANR RESYST)
Scientific team
Pierre Bras (Phd student), Thomas Dreyfus (CR CNRS), Andrew Elvey Price (CR CNRS) and Sofia Tarricone (post-doc)
Local department team (CITI, Institut Polytechnique de Paris)
Randal Douc (Professor), Yohan Petetin (Associate professor)
International conference funded by the RESYST project
Title: 40 years of reflected Brownian motion and related fields
Scientific committee: Irina Kurkova, Jean-François Le Gall, Michel Mandjes, Ruth Williams
Organizing committee: Sandro Franceschi, Héléne Guerin, Kilian Raschel
Date: April 24-28, 2023
Place: Roscoff in Brittany (France) in the CNRS conference center
Website: https://40yearsofrbm.wp.imt.fr/
Sponsors: Centre Henri Lebesgue, ANR JCJC grant « RESYST », ERC grant « Elliptic Combinatorics »
Spcial Issue of Queueing Systems
I am guest editor of the special issue Queueing Systems: Theory and Applications sur on Reflected Brownian Motion. This special issue is linked to the conference 40 years of reflected Brownian motion and related fields.
Call for papers and submissions process: https://www.springer.com/journal/11134/updates/23636094
Publications
HAL web page of the ANR project.
- The stationary distribution of reflected Brownian motion in a wedge: diffetential properties
Mireille Bousquet-Mélou, Andrew Elvey Price, Sandro Franceschi, Charlotte Hardouin, Kilian Raschel
Pdf HAL arXiv - Stationary Brownian motion in a 3/4-plane: reduction to a Riemann-Hilbert problem via Fourier transforms
Guy Fayolle, Sandro Franceschi, Kilian Raschel
Indagationes Mathematicae (2022) Special issue in honour of J. W. Cohen
Pdf HAL arXiv DOI - On the stationary distribution of reflected Brownian motion in a non-convex cone
Guy Fayolle, Sandro Franceschi, Kilian Raschel
Markov Processes and Related Fields (2022)
Pdf HAL arXiv DOI Animation GeoGebra - A dual skew symmetry for transient reflected Brownian motion in an orthant
Sandro Franceschi, Kilian Raschel
Queueing Systems (2022) Special issue in honour of Masakiyo Miyazawa
Pdf HAL arXiv DOI - Simulation of Reflected Brownian motion on two dimensional wedges
, Arturo Kohatsu-Higa
Pdf HAL arXiv DOI - Differential algebraic generating series of weighted walks in the quarter plane
Thomas Dreyfus
Pdf HAL arXiv DOI - Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones
Andrew Elvey Price
Pdf HAL arXiv DOI - Riemann-Hilbert approach to the generating function of the higher order Airy point processes
Mattia Cafasso, Sofia Tarricone
Pdf HAL arXiv DOI
Contact
Contact me if you have any questions !
sandro.franceschi@telecom-sudparis.eu
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