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Projet de Recherche

Multi-fragmentation processes related to rupture phenomena et EDP ​​non linéaires, bases de Lévy et mécanismes de branchement.

Part II: Nonlinear PDEs, Lévy bases, and branching mechanisms.

2022 - 2024 Equipe:.
  • Lucian Beznea - Institut de Mathématiques "Simion Stoilow" de l’ Académie Roumaine.
  • Oana Lupaşcu - Institut de Statistique Mathématique et de Mathématiques Appliqués "Gherghe Mihoc - Caius Iacob", Académie Roumaine.
  • Ioan R. Ionescu - Université Paris Nord.
  • Madalina Deaconu - INRIA et Université de Lorraine, Nancy,
  • Radu Stoica - Université de Lorraine, Nancy.
Activité 2022-2023:
  • Visite scientifique de Oana Lupascu-Stamate à l'Université Paris 13 pour collaboration avec Ioan R. Ionescu, 15-21 october 2023.
  • Visite scientifique de Oana Lupascu-Stamate et Lucian Beznea à l'Université Paris 13 pour collaboration avec Ioan R. Ionescu, 15-21 janvier 2023.
  • Visite scientifique de Oana Lupascu-Stamate à l'Université de Lorraine (Nancy) pour collaboration avec Madalina Deaconu, 30 juin - 9 julliet 2023.
  • Atelier de travail en Stochastiques et EDP, Video conference en ligne, Bucarest, 20-21 octobre 2020.
I. Branching multi-fragmentation model with interactions. We starting the inverstigation of the branching multi-fragmentation model, developed in [L. Beznea, I.R. Ionescu, and O. Lupascu-Stamate, J. of Evol. Equations 21 (2021)] by considering interactions/collisions between particles.
II. Branching processes and nonlinear PDEs. We investigated the two-dimensional (2d) vorticity equation that describes the time evolution of the vorticity of a fluid, namely the local rotation of the fluid in a bounded Euclidean domain. In the paper [S. Benachour, B. Roynette, P. Vallois, Rev. Mat. IberoAm. 17 (2001)] the solutions of the vorticity equation are represented by a stochastic model, through a branching process with state space the set of all finite configurations of the closure of the domain. A main tool is an appropriate branching processes as developed in [L. Beznea, O. Lupașcu-Stamate, C. Vrabie, Nonlinear Analysis 200 (2020)] and very recent stochastic numerical methods as in [B. Leimkuhler, A. Sharma, M.V. Tretyakov, Simplest random walk…, Preprint 2020].
III. Lévy bases with a branching mechanism. The Cox point processes represent one of the most important and versatile classes of point process models for clustered point patterns, see [G. Hellmund, M. Prokesova, E. B. Vedel Jensen, Adv. Appl. Prob. (SGSA) 40 (2008)]. We studied Cox point processes having driving intensity measures induced by a non-local branching mechanism, in particular, we studied the associated Choquet capacities.

Exposés:
  • Oana Lupascu-Stamate, Journées de Probabilités, Angers, juin 2023.
  • Oana Lupascu-Stamate , 1er Symposium de la recherche scientifique francophone en Roumanie , Bucarest, decembre 2022.
  • Oana Lupascu-Stamate , XV-eme Colloque Franco-Roumain de Mathematiques Appliquees, Toulouse, aout 2022.
  • Lucian Beznea , Journée PASTA, Nancy, julliet 2022.
  • Lucian Beznea , 1er Symposium de la recherche scientifique francophone en Roumanie , Bucarest, decembre 2022.
  • Lucian Beznea, Journée PASTA, Nancy, julliet 2022.

Part I: Multi-fragmentation processes related to rupture phenomena.

2018 - 2021
En coopération avec le Centre Francophone en Mathématiques de Bucarest (IMAR).
Equipe:.
  • Lucian Beznea - Institut de Mathématiques "Simion Stoilow" de l’ Académie Roumaine.
  • Oana Lupaşcu - Institut de Statistique Mathématique et de Mathématiques Appliqués "Gherghe Mihoc - Caius Iacob", Académie Roumaine.
  • Ioan R. Ionescu - Université Paris Nord.
Activité 2019:
  • Visite scientifique de Lucian Beznea et Oana Lupascu à l'Université Paris 13 pour collaboration avec Ioan R. Ionescu. 2 - 15 Fevrier 2020.
  • Séjour de Madalina Deaconu à IMAR Bucarest du 2 au 8 mars 2020. Collaboration avec Lucian Beznea et Oana Lupascu.
  • Visite scientifique de Ioan R. Ionescu (Universite Paris 13) à IMAR pour collaboration avec Lucian Beznea et Oana Lupascu. 24 Octobre – 4 Novembre 2019.
Activité 2018:
We studied fragmentation processes depending on several fragmentation kernels and on the position of the fragments in a given space, whith a spatial movement given by a (deterministic) right continuous flow, or equivalently, by a semi-dynamical system. The first aim was to complete this study by considering stochastic spatial movement, namely given by a fixed Markov process. The second aim was to understand the corresponding space of all (multi-fragmentation sizes in this situation, a convenient analogue of the space of J. Bertoin as considered in [Stoch. Proc. Appl. 25 (2015), 1861-1885] and [J. Stat. Phys. 162 (2016), 824-841], on which the forthcoming branching-fragmentation process will be constructed.

Exposé:
L. Beznea gave an invited talk on this subject at 14-ème Colloque Franco-Roumain de Mathématiques Appliquées, Bordeaux, August 2018.

Ateliers de travail à Bucarest:
  1. Atelier de travail Stochastique et EDP, Bucarest, (video-conférence en ligne) du 24 au 25 Janvier, 2019, <programme> <résumés>
  2. Atelier de travail Potentiel et Probabilit&eaciute;s, Bucarest, du 24 au 25 Janvier, 2019,
  3. Atelier de travail Théorie du potentiel et EDP non-linéaires,Bucarest, du 22 au 23 Novembre, 2018,
    Résumés
  4. Atelier de travail en stochastique et EDP, Bucarest, du 14 au 15 Septembre, 2018,
    Résumés
Publications:
  1. L. Beznea, I.R. Ionescu, O. Lupaşcu-Stamate, Random multiple-fragmentation and flow of particles on a surface, accepted to Journal of Evolution Equations, 2021.
  2. L. Beznea, M. Deaconu, O. Lupaşcu-Stamate, Scaling property for fragmentation processes related to avalanches. In: Applications of Mathematics and Informatics in Natural Sciences and Engineering, Springer Proceedings in Mathematics &Statistics 334 (2020),
  3. L. Beznea, O. Lupașcu-Stamate, C. Vrabie, Stochastic solutions to evolution equations of non-local branching processes, Nonlinear Analysis 200 (2020), 112021, 18 pp.
  4. M. Deaconu, O. Lupascu-Stamate, Asymptotic behaviour of a one-dimensional avalanche model through a particular stochastic process,preprint 2023.
  5. L. Beznea, O. Lupaşcu-Stamate, Radu Stoica, Levy driven branching Cox processes, preprint 2024.
  6. I. R. Ionescu, O. Lupascu-Stamate, Boundary variation method for the generalized Cheeger problem, Applied Numerical Mathematics 140 (2019) 199-214.
  7. L. Beznea, M. Deaconu, O. Lupascu-Stamate, “Numerical approach for stochastic differential equations of fragmentation; application to avalanches”, Mathematics and Computers in Simulation, 160 (2019), 111--125.;
  8. L. Beznea, I.R. Ionescu, and O. Lupascu-Stamate, “Multiple-fragmentation driven by a spatial Markov process”, work in preparation for publication.
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