Research page of Matthias Winkel

Contact details

Email: winkel@stats.ox.ac.uk
Phone: +44 1865 2 81886
Postal address: Department of Statistics, 1 South Parks Road, Oxford OX1 3TG, UK
URL: http://www.stats.ox.ac.uk/~winkel/indexe.html

Research interests: stochastic processes and applications

Main interests

Discrete and continuum random trees and forests, branching processes, superprocesses, tree-valued random processes
Exchangeability, random partitions, compositions, random hierarchies, interval partitions
Lévy processes, subordinators, time changes

Current Ph.D. students

Matthew Buckland (2020--)
David Geldbach (2021--, joint supervision with Julien Berestycki)

I'm always happy to consider prospective Ph.D. students interested in my main research interests. Feel free to send me an email if you'd like to find out more about what's behind these topics. Briefly, my aim is to follow a probabilistic approach sometimes starting from heuristic arguments, but some advanced probability theory and analytical tools are required to then establish results rigorously. This is not meant to sound like a hidden list of prerequisites but you should expect to spend a good proportion of your time, particularly initially, doing guided reading of selected parts of the literature to extend your tool kit.

Research environment

I am a member of the Probability Group.
Other related research groups: Stochastic Analysis, Combinatorics

Collaborators and other contacts

Collaborators: Jim Pitman (Berkeley), Ole Barndorff-Nielsen (Aarhus), Neil Shephard (Harvard), Thomas Duquesne (Paris 6), Bénédicte Haas (Université Sorbonne Paris Nord), Grégory Miermont (Lyon), Qingyang Guan (Chinese Academy of Sciences, Beijing), Peter McCullagh (Chicago), Daniel Ford (then at Google), Almut Veraart (Imperial College London), Mladen Savov (University of Sofia), Douglas Rizzolo (University of Delaware), Maren Eckhoff (then at QuantumBlack), Andreas Kyprianou (Bath), Hui He (Beijing Normal University), Soumik Pal (University of Washington, Seattle), Noah Forman (McMaster University, Hamilton)
Former Ph.D. students: Almut Veraart (jointly supervised with Neil Shephard, completed in 2007, now at Imperial College London), Di Kuang (jointly supervised with Bent Nielsen, completed in 2008, then at Aon), Bo Chen (completed in 2009, then Goldman Sachs), John Cao (completed in 2011, then Royal Bank of Scotland), Patrick Hoscheit (visiting in 2011, now at INRA), Franz Rembart (completed in 2016, now at Boston Consulting and Columbia University), Dane Rogers (completed in 2020, then medical statistician), Frederik Soerensen (completed in 2021, then BASE life science)

Publications with links to pdf files (preprint versions, where appropriate)

Winkel, M.: Burgers turbulence initialized by a regenerative impulse. Stoch. Proc. Appl. 93, 241-268 (2001)
Winkel, M.: Right inverses of non-symmetric Lévy processes. Ann. Prob. 30, 382-415 (2002)
Winkel, M.: Limit clusters in the inviscid Burgers turbulence with certain random initial velocities. J. Stat. Phys. 107, 893-917 (2002)
Winkel, M.: Electronic foreign exchange markets and passage events of independent subordinators. J. Appl. Prob. 42, 138-152 (2005)
Winkel, M.: The recovery problem for time-changed Lévy processes. Preprint
Pitman, J. and Winkel, M.: Growth of the Brownian forest. Ann. Prob. 33, 2188-2211 (2005)
Barndorff-Nielsen, O.E., Shephard, N. and Winkel, M.: Limit theorems for multipower variation in the presence of jumps. Stoch. Proc. Appl. 116, 796-806 (2006)
Duquesne, T. and Winkel, M.: Growth of Lévy trees. Probability Theory and Related Fields 139, no. 3-4, 313-371 (2007)
Haas, B., Miermont, G., Pitman, J. and Winkel, M.: Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Prob. 36, No. 5, 1790-1837 (2008)
Guan, Q.-Y. and Winkel, M.: SLE and alpha-SLE driven by Lévy processes. Ann. Prob. 36, No. 4, 1221-1266 (2008)
McCullagh, P., Pitman, J. and Winkel, M.: Gibbs fragmentation trees. Bernoulli 14(4), 988-1002 (2008)
Haas, B., Pitman, J. and Winkel, M.: Spinal partitions and invariance under re-rooting of continuum random trees.. Ann. Prob. 37, No. 4, 1381-1411 (2009)
Pitman, J. and Winkel, M.: Regenerative tree growth: binary self-similar continuum random trees and Poisson-Dirichlet compositions. Ann. Prob. 37, No. 5, 1999-2041 (2009)
Chen, B., Ford, D. and Winkel, M.: A new family of Markov branching trees: the alpha-gamma model. Elec. J. Prob. 14, 400-430 (2009)
Veraart, A. and Winkel, M.: Time change. Encyclopedia of Quantitative Finance, ed. R. Cont, Wiley (2010)
Savov, M. and Winkel, M.: Right inverses of Lévy processes: the excursion measure in the general case. Elec. C. Prob. 15, 572-584 (2010)
Cao, X. and Winkel, M.: Growth of Galton-Watson trees: immigration and lifetimes. arXiv:1004.3061 Preprint (2010)
Duquesne, T. and Winkel, M.: Hereditary tree growth and Lévy forests. Stoch. Proc. Appl. 129, 3690-3747 (2019)
Chen, B. and Winkel, M.: Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees. Ann. Inst. H. Poincaré Vol. 49, No. 3, 839-872 (2013)
Pitman, J., Rizzolo, D. and Winkel, M.: Regenerative tree growth: structural results and convergence. Electron. J. Probab. 19 (2014), no.70, 1--27
He, H. and Winkel, M.: Invariance principles for pruning processes of Galton-Watson trees. arXiv:1409.1014 (2014)
Eckhoff, M., Kyprianou, A. E. and Winkel, M.: Spines, skeletons and the Strong Law of Large Numbers for superdiffusions. Ann. Prob. 43, No. 5, 2594-2659 (2015)
Pitman, J. and Winkel, M.: Regenerative tree growth: Markovian embedding of fragmenters, bifurcators and bead splitting processes. Ann. Prob. 43, No. 5, 2611-2646 (2015)
Rembart, F. and Winkel, M.: Recursive construction of continuum random trees. Ann. Probab. 46, No. 5, 2715-2748 (2018)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Uniform control of local times of spectrally positive stable processes. Ann. Appl. Probab. 28, No. 4, 2592-2634 (2018)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Diffusions on a space of interval partitions with Poisson-Dirichlet stationary distributions. arXiv:1609.06706 (2016) superseded by four papers below
Rembart, F. and Winkel, M.: A binary embedding of the stable line-breaking construction. Stoch. Process. Appl. 163 (2023) 424-472
He, Hui and Winkel, M.: Gromov-Hausdorff-Prokhorov convergence of vertex cut-trees of n-leaf Galton-Watson trees. arXiv:1704.00044, Bernoulli 25, 2301-2329 (2019)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Projections of the Aldous chain on binary trees: intertwining and consistency. Random Struct. Alg. 57(3):745-769 (2020)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Interval partition evolutions with emigration related to the Aldous diffusion. arXiv:1804.01205 (2018) superseded by monograph on the Aldous diffusion below
Pitman, J. and Winkel, M.: Squared Bessel processes of positive and negative dimension embedded in Brownian local times. Elec. Comm. Probab. 23 (2018), paper no. 74
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Aldous diffusion I: a projective system of continuum k-tree evolutions. arXiv:1809.07756 (2018) superseded by monograph on the Aldous diffusion below
Chee, N., Rembart, F. and Winkel, M.: A recursive distribution equation for the stable tree. arxiv:1812.08636 (2018), to appear in Bernoulli (2023)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Metrics on sets of interval partitions with diversity. Electron. Commun. Probab. 25:16 pp. (2020)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Diffusions on a space of interval partitions: construction from marked L\'evy processes. Electron. J. Probab., 25: 45pp. (2020) Supplement, Simulation
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Diffusions on a space of interval partitions: Poisson-Dirichlet stationary distributions. Ann. Probab., 49(2):793-831 (2021)
Winkel, M.: Diffusions on a space of interval partitions: construction from Bertoin's BES_0(d), 0\le d\le 1. Electron. Commun. Probab., 25:13pp. (2020)
Rogers, D. and Winkel, M.: A Ray-Knight representation of up-down Chinese restaurants. Bernoulli 28(1), 689-712 (2022)
Forman, N., Rizzolo, D., Shi, Q. and Winkel, M.: A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions. Ann. Appl. Probab. 32(3):2211-2253 (2022)
Forman, N., Rizzolo, D., Shi, Q. and Winkel, M.: Diffusions on a space of interval partitions: the two-parameter model. Electron. J. Probab., 28: 46pp. (2023)
Shi, Q. and Winkel, M.: Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions. ALEA, Lat. Am. J. Probab. Math. Stat. 20, 665-696 (2023)
Shi, Q. and Winkel, M.: Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits arXiv:2012.15758 (2020)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: Ranked masses in two-parameter Fleming-Viot diffusions. Trans. Amer. Math. Soc. 376(2): 1089-1111 (2023)
Forman, N., Pal, S., Rizzolo, D. and Winkel, M.: The Aldous diffusion: a stationary evolution of the Brownian CRT. arXiv:2305.17269 (2023)

Brief version of my curriculum vitae

M.Sc. (Manchester, 1998, supervisor Ron Doney), Diplom (Münster, 1999, supervisor Gerold Alsmeyer), Ph.D. (Paris, 2001, supervisor Jean Bertoin (now Zurich))
04/01-03/02 Pre- and postdoctoral stay at the University of Aarhus, supervisor: Ole Barndorff-Nielsen
since 04/02 Departmental Lecturership, since 03/17 Research Lecturership, since 03/18 title of Associate Professor at the Department of Statistics, University of Oxford,
10/02-12/06 Associate Membership at Nuffield College Oxford, 04/07-09/09 College Lecturership at St Catherine's College Oxford, 01/11-09/12 College Lecturership at Somerville College Oxford, 02/03-09/13 College Lecturership at Trinity College Oxford,
since 10/12 College Lecturership at Brasenose College Oxford, since 01/19 Supernumerary Fellowship