Publications

 Preprints

• Convergence rates for the Adam optimizer
  joint work with Arnulf Jentzen
  arXiv:2407.21078
  Preprint

• Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates
  joint work with Robin Graeber and Arnulf Jentzen
  arXiv:2407.08100
  Preprint

• Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses
  joint work with Arnulf Jentzen and Adrian Riekert
  arXiv:2406.14340
  Preprint

• Traces left by random walks on large random graphs: local limits
  arXiv:2403.14787
  Preprint

• On the existence of optimal shallow feedforward networks with ReLU activation
  joint work with Sebastian Kassing
  J. Mach. Learn. 3, No. 1, 1-22 (2024), DOI: 10.4208/jml.230903
  PDF

• On the existence of minimizers in shallow residual ReLU neural network optimization landscapes
  joint work with Arnulf Jentzen and Sebastian Kassing
  to appear in SINUM, DOI: 10.1137/23M1556241
  Preprint
• Quasi-processes for branching Markov chains
  joint work with Martin Maiwald
  arXiv:2107.06654
  Preprint
• Convergence of stochastic gradient descent schemes for Lojasiewicz-landscapes
  joint work with Sebastian Kassing
  J. Mach. Learn. 3, No. 3, 245-281 (2024), DOI: 10.4208/jml.240109
  PDF

Artificial neural networks / Stochastic approximation

• Central limit theorems for stochastic gradient descent with averaging for stable manifolds
  joint work with Sebastian Kassing
  Electron. J. Probab. 28: 1-48 (2023), DOI: 10.1214/23-EJP947
  PDF
• On minimal representations of shallow ReLU networks
  joint work with Sebastian Kassing
  Neural Networks 148, 121-128 (2022), DOI 10.1016/j.neunet.2022.01.006
  Preprint
• Cooling down stochastic differential equations: almost sure convergence
  joint work with Sebastian Kassing
  Stochastic Process. Appl. 152, 289-311 (2022), DOI 10.1016/j.spa.2022.06.020
  Preprint
• General multilevel adaptations for stochastic approximation algorithms II: CLTs
  Stochastic Process. Appl. 132, 226-260 (2021), DOI 10.1016/j.spa.2020.11.001
  Preprint
• General multilevel adaptations for stochastic approximation algorithms of Robbins-Monro and Polyak-Ruppert type
  joint work with Thomas Müller-Gronbach
  Numer. Math. 142, No. 2, 279-328 (2019), DOI 10.1007/s00211-019-01024-y
  Preprint

Complex networks / Condensation

• The rank-one and the preferential attachment paradigm
  In Network Science - An Aerial View, 43-58 (2019)
• Local neighbourhoods for first passage percolation on the configuration model
  joint work with Marcel Ortgiese
  J. Stat. Phys. 137, No. 3-4, 485-501 (2018), DOI 10.1007/s10955-018-2028-7
  Preprint
• The shape of the emerging condensate in effective models of condensation
  joint work with Volker Betz and Peter Mörters
  Ann. Henri Poincaré 19, No. 6, 1869-1889 (2018)
  Preprint
• Non-extensive condensation in reinforced branching processes
  joint work with Cécile Mailler and Peter Mörters
  Ann. Appl. Probab. 27, No. 4, 2539-2568 (2017)
  Preprint
• Distances in scale free networks at criticality
  joint work with Christian Mönch and Peter Mörters
  Electron. J. Probab. 22, No. 77, 38 pp. (2017)
  PDF
• Preferential attachment with fitness: unfolding the condensate
  Electron. J. Probab. 21, No. 3, 1-38 (2016)
  PDF
• Cycle length distributions in random permutations with diverging cycle weights
  joint work with Peter Mörters
  Random. Struct. Algor. 46, No. 4, 635-650 (2015)
  PDF
• Robust analysis of preferential attachment models with fitness
  joint work with Marcel Ortgiese
  Combin. Probab. Comput. 23, No. 3, 386-411 (2014), DOI 10.1017/S0963548314000157
  Preprint
• Emergence of condensation in Kingman's model of selection and mutation
  joint work with Peter Mörters
  Acta Appl. Math. 127, No. 1, 17-26 (2013), DOI 10.1007/s10440-012-9790-3 
  Preprint 
• Typical distances in ultrasmall random networks
  joint work with Christian Mönch and Peter Mörters
  Adv. in Appl. Probab. 44, No. 2, 583-601 (2012) 
  Preprint
• Random networks with sublinear preferential attachment: the giant component
  joint work with Peter Mörters
  Ann. Probab. 41, No. 1, 329-384 (2013), DOI 10.1214/11-AOP697
  Preprint
• Random Networks with Concave Preferential Attachment Rule
  joint work with Peter Mörters
  Jahresber Dtsch Math-Ver 113, No. 1, 21-40 (2011), DOI 10.1365/s13291-010-0011-6
• Random networks with sublinear preferential attachment: Degree evolutions
  joint work with Peter Mörters
  Electron. J. Probab. 14, 1222-1267 (2009)
  PDF

Numerical integration / Quadrature of SDEs

• Multilevel Monte Carlo for Lévy-driven SDEs: central limit theorems for adaptive Euler schemes
  joint work with Sangmeng Li
  Ann. Appl. Probab. 26, No. 1, 136-185 (2016)
  PDF
• Quadrature for self-affine distributions on R^d
  joint work with Thomas Müller-Gronbach
  Found. Comput. Math. 15, No. 6, 1465-1500 (2015), DOI 10.1007/s10208-014-9233-9
  Preprint
• On the complexity of computing quadrature formulas for SDEs
  joint work with Thomas Müller-Gronbach and Klaus Ritter
  in Foundations of Computational Mathematics, Budapest 2011 (eds. F. Cucker, T. Krick, A. Pinkus, A. Szanto), LMS Lecture Notes 403, Cambridge University Press, 2012. DOI 10.1017/CBO9781139095402
  Preprint 
• An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process
  joint work with Andreas Neuenkirch and Lukas Szpruch
  Proc. R. Soc. A 468, No. 2140, 1105�1115 (2012), DOI 10.1098/rspa.2011.0505
  Preprint
• Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction
  Ann. Appl. Probab. 21, No. 1, 283-311 (2011)
  Preprint
• A multilevel Monte Carlo algorithm for Lévy driven stochastic differential equations
  joint work with Felix Heidenreich
  Stochastic Process. Appl. 121, Mo. 7, 1565-1587 (2011)
  Preprint
• Infinite-dimensional quadrature and approximation of distributions
  joint work with Jakob Creutzig, Thomas Müller-Gronbach and Klaus Ritter
  Found. Comput. Math. 9, No. 4, 391-429 (2009)
  Preprint

Small deviations and approximation theory (quantization)

• Persistence probabilities for an integrated random walk bridge
  joint work with Frank Aurzada and Mikhail Lifshits
  Probab. Math. Statist. 34, No. 1, 1-22 (2014)
  Preprint
• Constructive quantization: approximation by empirical measures
  joint work with Michael Scheutzow and Reik Schottstedt
  Ann. Inst. Henri Poincaré (B) 49, No. 4, 1183-1203 (2013)
  Preprint
• Universality of the asymptotics of the one-sided exit problem for integrated processes
  joint work with Frank Aurzada
  Ann. Inst. Henri Poincaré (B) 49, No. 1, 236-251 (2013), DOI 10.1214/11-AIHP427 
  Preprint
• The high resolution vector quantization problem with Orlicz norm distortion
  joint work with Christian Vormoor
  J. Theor. Probab. 24, No. 2, 517-544 (2011)
  Preprint
• Small deviations of general Lévy processes
  joint work with Frank Aurzada
  Ann. Probab. 37, No.5, 2066-2092 (2009)
  Preprint
• High resolution quantization and entropy coding of jump processes
  joint work with Frank Aurzada, Michael Scheutzow and Christian Vormoor
  J. Complexity 25, 163-187 (2009)
  Preprint
• The coding complexity of Lévy processes
  joint work with Frank Aurzada
  Found. Comput. Math. 9, No. 3, 359-390 (2009)
  Preprint
• Asymptotic formulae for coding problems and intermediate optimization problems: a review
  in Trends in Stochastic Analysis, edited by Jochen Blath, Peter Morters, and Michael Scheutzow, Cambridge University Press, 187-232, 2009
  Preprint
• The coding complexity of diffusion processes under supremum norm distortion
  Stochastic Process. Appl. 118, No.6, 917-937 (2008), doi:10.1016/j.spa.2007.07.003
  Preprint
• The coding complexity of diffusion processes under L^p[0,1]-norm distortion
  Stochastic Process. Appl. 118, No.6, 938-951 (2008), doi:10.1016/j.spa.2007.07.002
  Preprint
• High-resolution quantization and entropy coding for fractional Brownian motion
  joint work with Michael Scheutzow
  Electron. J. Probab. 11, 700-722 (2006)
  Preprint
• Probabilities of randomly centered small balls and quantization in Banach spaces
  joint work with Mikhail A. Lifshits
  Ann. Probab. 33, No.4, 1397-1421 (2005)
  PDF
• Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces
  Bulletin des Sciences Mathématiques 129, No.10, 791-803 (2005)
• Small ball probabilities around random centers of Gaussian measures and applications to quantization
  J. Theor. Probab. 16, No.2, 427-449 (2003)
• On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces
  joint work with Franz Fehringer, Anis Matoussi and Michael Scheutzow
  J. Theor. Probab. 16, No.1, 249-265 (2003)

Stochastic analysis

• Real self-similar processes started from the origin
  joint work with Leif Döring and Andreas E. Kyprianou
  Ann. Probab. 45, No. 3, 1952-2003 (2017) 
  Preprint
• A support theorem and a large deviation principle for Kunita flows
  joint work with Georgi Dimitroff
  Stochastics and Dynamics 12, No. 3 (2012)
  Preprint
• Rough paths analysis of general Banach space-valued Wiener processes
  J. Funct. Anal. 258, No. 9, 2910-2936 (2010)
  Preprint
• Enlargement of filtrations and continuous Girsanov-type embeddings
  joint work with Stefan Ankirchner and Peter Imkeller
  in Séminaire de Probabilités XL (ed. Catherine Donati-Martin, Michel Émery, Alain Rouault, and Christophe Stricker), LNM1899, Springer, 2007
• The Shannon information of filtrations and the additional logarithmic utility of insiders
  joint work with Stefan Ankirchner and Peter Imkeller
  Ann. Probab. 34, No.2, 743-778 (2006)
  PDF

PhD Thesis

• High resolution coding of stochastic processes and small ball probabilities
  Ph.D. Dissertation (2003)
  PDF