Themes for Learning Sessions

This is an overview of themes for the Learning Sessions (31 July - 4 August 2017) that will be part of the program Genealogies of Interacting Particle Systems at the Institute for Mathematical Sciences of the National University of Singapore. Based on the preferences of participants, we have chosen nine learning sessions that will be organized. The intended format of the Learning Sessions is briefly explained on the webpages of the program. Two people (moderators) together prepare an introduction to the topic in the form of an introductory lecture of (roughly) 90 minutes and formulate questions and problems to be discussed in the session. Moderators have a genuine interest in learning the topic but are not themselves authors of the papers to be studied.

Program

Some information about the sessions

• The Continuum Random Tree Moderators: Anita Winter and Emmanuel Schertzer. Based on:
  • Pitman, Enumerations of trees and forests related to branching processes and random walks, Microsurveys in discrete probability (1997) (for a proof that particular GW-trees conditioned on number of vertices are uniform ordered/binary/unordered, labelled trees = OtterDwass-Formel)
  • Aldous, Continuum random tree III, AoP 1993, (for the definition and a proof of convergence of suitably rescaled GW trees conditioned on the number of vertices towards the CRT)
  • Marckert & Mokkadem, The depth first process of Galton-Watson trees converge to the same Brownian excursion, AoP, 2003 (for another proof which uses Contour functions, Height function and the Lukaciewiec walk)
  • Bennies & Kersting, A random walk approach to Galton-Watson trees, Journal of theoretical probability, 2000 (alternative to the previous paper, but slightly weaker)
  • Bertoin & Miermont, The cut-tree of large Galton-Watson trees and the Brownian CRT, AoP 2013 (traces back the genealogy of parition elements of the CRT obtained by letting rain down cut points/mutations on the CRT; proof makes use of the additive coalescent) and shows that the cut tree is again a CRT)
  • David Aldous (1997). Brownian excursions, critical random graphs and the multiplicative coalescent, Annals of probsability.
  • Louigi Addario-Berry, Nicolas Broutin and Christina Goldschmidt, The continuum limit of critical random graphs, Probability Theory and Related Fields
  • David Aldous and Jim Pitman (1998). The standard additive coalescent. Annals of Probability.
• Tree-valued Markov processes Moderators: Andrej Depperschmidt and Andreas Greven. The description of processes via flow of bridges has been created in three papers of Bertoin and Le Gall 2003, 2005, 2006:
  • [1] Bertoin, Jean; Le Gall, Jean-François. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003), no. 2, 261–288.
  • [2] Bertoin, Jean; Le Gall, Jean-François. Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 307–333.
  • [3] Bertoin, Jean; Le Gall, Jean-Francois. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006), no. 1-4, 147–181.
  This approach has been generalized and unified by Dawson and Li 2012
  • [4] Dawson, Donald A.; Li, Zenghu. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012), no. 2, 813–857.
  and subsequently a lot of applications where developed:
  • [5] Labbé, Cyril. Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 3, 732–769.
  • [6] Foucart, Clément. Generalized Fleming-Viot processes with immigration via stochastic flows of partitions. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), no. 2, 451–472.
  In particular there is some work which gives a connection with the session on Lookdown constructions.
  • [7] Stephan Gufler. Pathwise construction of tree-valued Fleming-Viot processes arXiv:1404.3682v3
  The program of the session is as follows:
  • Moderators: Introduction into the main ideas of the basic references 1-4, (90 minutes).
  • M.Birkner, A.Blancas-Benitez: The more recent developments references 5-7 (40 minutes).
  • Structured but open discussion: Possible further applications and extensions (30 minutes). [Some suggestions by moderators/participants.]
• Look down constructions Moderators: Amandine Véber and Anton Wakolbinger. Based, e.g., on:
  • P. Donnelly and T. Kurtz (1999), Particle representations for measure-valued population models, Ann. Probab. 27 , no. 1, 166–205.
  • T. Kurtz and E. Rodrigues (2011),, Poisson representations of branching Markov and measure-valued branching processes, Ann. Probab. 39 939-984.
  • A. Etheridge and T. Kurtz (2016), Genealogical constructions of population models
• Genealogies of particles on dynamic random networks Moderators: Anton Klimovsky and Jiří Černý. Motivated, e.g., by complex network modelling and statistical issues like inference of network model parameters. One could try to learn more about graph limits, sparse graphs vs. exchangeability, dynamic random network models (graph-valued Markov processes), and explore e.g., open questions on particle systems in the spirit of Aldous' "Interacting particle systems as stochastic social dynamics" in the context of time-varying underlying network models. Based on, e.g.,
  • Aldous, David. "Interacting particle systems as stochastic social dynamics." Bernoulli 19.4 (2013): 1122-1149.
  • Crane, Harry. "Time-varying network models." Bernoulli 21.3 (2015): 1670-1696.
  • Crane, Harry. "Dynamic random networks and their graph limits." The Annals of Applied Probability 26.2 (2016): 691-721.
  • Borgs, Christian, et al. "Sparse exchangeable graphs and their limits via graphon processes." arXiv preprint arXiv:1601.07134 (2016).
  • Avena, Luca, et al. "Mixing times of random walks on dynamic configuration models." arXiv preprint arXiv:1606.07639 (2016).
• The Brownian web and net Moderators: Nic Freeman and Sandra Palau. Based, e.g., on:
  • E. Schertzer, R. Sun and J.M. Swart. The Brownian web, the Brownian net, and their universality. (arXiv:1506.00724) Advances in Disordered Systems, Random Processes and Some Applications, 270--368, Cambridge University Press, 2017.
  • L.R.G. Fontes, M. Isopi, C.M. Newman, K. Ravishankar. The Brownian web: characterization and convergence. Ann. Probab. 32(4), 2857{2883, 2004.
  • R. Sun and J.M. Swart. The Brownian net. Ann. Probab. 36, 1153-1208, 2008.
  • C.M. Newman, K. Ravishankar, and E. Schertzer. Marking (1,2) points of the Brownian web and applications. Ann. Inst. Henri Poincaré Probab. Statist. 46, 537-574, 2010.
  • E. Schertzer, R. Sun and J.M. Swart (2010): Stochastic flows in the Brownian web and net. (Mem. Am. Math. Soc. Vol. 227 (2013), Nr. 1065)
• Extremal particles in Branching Brownian Motion Moderators: Sandra Kliem and Kumarjit Saha. This session will be based on the following articles & lecture notes:
  • J. Berestycki, N. Berestycki, and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Ann. Probab. 41(2) (2013), 527–618.
  • Arguin, Bovier, and Kistler. The genealogy of extremal particles of Branching Brownian Motion. CPAM, 64, (2011)
  • Arguin, Bovier, and Kistler. Poissonian statistics in the extremal process of BBM, Annals of Applied Probability 2012, Vol 22, No. 4
  • A. Bovier. From spin glasses to branching Brownian motion--and back? (Lecture notes for a course given at the 7th Prague Summer School on Mathematical Statistical Mechanics, August 19--30, 2013).
  The session will be divided into two parts: Branching Brownian Motion without selection (Sandra Kliem) and with selection (Kumarjit Saha). The second part uses the following: Lecture Notes Branching Brownian Motion with Selection
• Competing species models Moderators: Matthias Hammer and Yu-Ting Chen. Based, e.g., on:
  • C. Neuhauser and S.W. Pacala. An explicitly spatial version of the Lotka–Volterra model with interspecific competition. Ann. Appl. Probab. 9, (1999), 1226–1259.
  • J. Blath, A.M. Etheridge and M.E. Meredith. Coexistence in locally regulated competing populations and survival of BARW. Ann. Appl. Probab. 17, (2007), 1474–1507.
  • A. Sturm and J.M. Swart (2007): Voter models with heterozygosity selection. (Ann. Appl. Probab. 18 No. 1 (2008), 59-99)
• Dualities in population models Moderators: Daniel Valesin, Fernando Cordero, and Stein Andreas Bethuelsen. In this learning session we will focus on graphical representations of dualities and pathwise duality. Our presentation will mainly be based on the following two papers:
  • M. Hutzenthaler and R. Alkemper. Graphical representation of some duality relations in stochastic population models. Electron. Commun. Probab. 12 (2007), 206–220.
  • A. Sturm and J.M. Swart. Pathwise duals of monotone and additive Markov processes. J. Theor. Probab. (2016). doi:10.1007/s10959-016-0721-5. 52 pages.
  We might also use material from related papers, such as:
  • J.M. Swart. Duals and thinnings of some relatives of the contact process. (2006) (18 pages) arXiv:math.PR/0604335.
  • S. Jansen and N. Kurt. Graphical representation of certain moment dualities and application to population models with balancing selection. Electron. Commun. Probab. 18 (2013), 1-15.
  In this learning session we will present the main ideas from the above papers. People interested in the session are advised to read the above papers in advance, although it is not absolutely necessary.
• The algebraic approach to duality Moderators: Anja Sturm, Florian Völlering, and Jan Swart. We used the following Lecture Notes, here an updated version, which are in turn based on the following papers:
  • C. Giardinà, J. Kurchan, F. Redig, K. Vafayi. Duality and hidden symmetries in interacting particle systems. Journal of Statistical Physics 135(1), (2009), 25–55.
  • G. Carinci, C. Giardinà, C. Giberti, F. Redig. Dualities in population genetics: a fresh look with new dualities. Stochastic Processes Appl. 125(3), (2015), 941-969.
  • A. Sudbury and P. Lloyd. Quantum operators in classical probability theory. II: The concept of duality in interacting particle systems. Ann. Probab. 23(4) (1995), 1816-1830.
  • A. Sudbury. Dual families of interacting particle systems on graphs. J. Theor. Probab. 13(3), (2000), 695-716.
  We also used Chapters 1-4 of
  • B.C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics, vol. 222, Springer, 2003.
  which contains standard facts about representations of Lie algebras that mathematicians and theoretical physicists are supposed to know.