Prof. Dr. Frank Aurzada Technische Universität Darmstadt Fachbereich Mathematik Arbeitsgruppe Stochastik Schlossgartenstraße 7 64289 Darmstadt Germany Email: aurzada [at] mathematik [dot] tu-darmstadt [dot] de Phone: +49 (0) 6151 - 16 - 23375 Office: S2|15  Room: 341

Teaching and events

• Summer 2024: sabbatical
  
  
• Office hours by appointment via e-mail
   
• Spring School "Multiplicative chaos and cascades", February 19-23, 2024
   
• conferences and schools organized in the past
   

Publications

• Brownian motion conditioned to spend limited time outside a bounded interval - an extreme example of entropic repulsion.
  submitted
  joint work with Martin Kolb (Paderborn) and Dominic T. Schickentanz (Darmstadt)
   
• The Best Time for an Update: Risk-Sensitive Minimization of Age-Based Metrics.
  submitted
  joint work with Wanja de Sombre, Andrea Ortiz, and Anja Klein (Darmstadt)
   
   
   
• Brownian motion conditioned to have restricted L_2-norm.
  submitted
  joint work with Mikhail Lifshits (St. Petersburg) and Dominic T. Schickentanz (Darmstadt)
   
• Time- versus Event-Triggered Consensus for a Single-Integrator Multi-Agent System.
  Nonlinear Analysis: Hybrid Systems 53 (2024), 101494
  joint work of David Meister (Stuttgart), Frank Aurzada, Mikhail Lifshits (St. Petersburg), and Frank Allgöwer (Stuttgart)
   
• Persistence probabilities of a smooth self-similar anomalous diffusion process.
  Journal of Statistical Physics 191 (2024), no. 37.
  joint work with Pascal Mittenbühler (Paderborn)
   
• Scaling limit of stretched Brownian chains.
  Journal of Physics A: Mathematical and Theoretical 56 (2023), 435006
  joint work with Volker Betz (Darmstadt) and Mikhail Lifshits (St. Petersburg)
   
• Persistence probabilities of weighted sums of stationary Gaussian sequences.
  Stochastic Processes and their Applications 159 (2023), 286-319.
  joint work with Sumit Mukherjee (Columbia University)
   
• Persistence probabilities of mixed FBM and other mixed processes.
  Journal of Physics A: Mathematical and Theoretical, 55 (2022) 305003.
  joint work with Martin Kilian (Darmstadt) and Ercan Sönmez (Klagenfurt)
   
• Brownian motion conditioned to spend limited time below a barrier.
  Stochastic Processes and their Applications 146 (2022), 360-381.
  joint work with Dominic T. Schickentanz (Darmstadt)
   
• Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion.
  Theory of Probability and Its Applications 67 (2022), 77-88 (English).
  Teoriya Veroyatnostei i ee Primeneniya 67 (2021), 100-114 (Russian).
  joint work with Martin Kilian (Darmstadt)
   
• Breaking a chain of interacting Brownian particles.
  Annals of Applied Probability 31 (2021), 2585-2611.
  joint work with Volker Betz (Darmstadt) and Mikhail Lifshits (St. Petersburg)
   
• Breaking a chain of interacting Brownian particles: A Gumbel limit theorem.
  Theory of Probability and Its Applications 66 (2021), 84-208 (English).
  Teoriya Veroyatnostei i ee Primeneniya 66 (2021), 231-260 (Russian).
  joint work with Volker Betz (Darmstadt) and Mikhail Lifshits (St. Petersburg)
   
• Persistence exponents in Markov chains.
  Annales de l'Institut Henri Poincaré - Probabilités et Statistiques 57 (2021), 1411-1441.
  joint work with Sumit Mukherjee (Columbia University) and Ofer Zeitouni (Weizmann Institute)
   
• Universal break law for chains of Brownian particles with nearest neighbour interaction.
  Journal of Physics A: Mathematical and Theoretical 54 (2021) 305204.
  joint work with Volker Betz (Darmstadt) and Mikhail Lifshits (St. Petersburg)
   
• Penalizing fractional Brownian motion for being negative.
  Stochastic Processes and their Applications 130 (2020), 6625-6637.
  joint work with Micha Buck and Martin Kilian (Darmstadt)
   
• Ruin probabilities in the Cramér-Lundberg model with temporarily negative capital.
  European Actuarial Journal 10 (2020), 261-269.
  joint work with Micha Buck (Darmstadt)
   
• Large deviations for infinite weighted sums of stretched exponential random variables.
  Journal of Mathematical Analysis and Applications 485 (2020), 123814.
   
• Persistence exponents via perturbation theory: AR(1)-processes.
  Journal of Statistical Physics 177 (2019), 651--665.
  joint work with Marvin Kettner (Darmstadt)
   
• Scaling limits for a random boxes model.
  Advances in Applied Probability 51 (2019), 773-801.
  joint work with Sebastian Schwinn (Darmstadt)
   
• The first exit time of fractional Brownian motion from a parabolic domain.
  Theory of Probability and Its Applications 64 (2019), 490-497 (English).
  Teoriya Veroyatnostei i ee Primeneniya 64 (2019), 610-620 (Russian).
  joint work with Mikhail Lifshits (St. Petersburg)
   
• How complex is a random picture?
  Journal of Complexity 53 (2019), 133-161.
  joint work with Mikhail Lifshits (St. Petersburg)
   
• Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes.
  Theory of Probability and Its Applications 63 (2019), 664-670 (English).
  Teoriya Veroyatnostei i ee Primeneniya 63 (2018), 817-826 (Russian).
  joint work with Christian Mönch (Darmstadt)
   
• Improving the performance of polling models using forced idle times.
  Probability in the Engineering and Informational Sciences 32 (2018), 580-602.
  joint work with Sebastian Schwinn (Darmstadt)
   
• Persistence probabilities of two-sided (integrated) sums of correlated stationary Gaussian sequences.
  Journal of Statistical Physics 170 (2018), 784-799.
  joint work with Micha Buck (Darmstadt)
   
• Persistence probabilities for stationary increment processes.
  Stochastic Processes and their Applications 128 (2018), 1750-1771.
  joint work with Nadine Guillotin-Plantard (Lyon) and Françoise Pène (Brest)
   
• Random walks and branching processes in correlated Gaussian environment.
  Journal of Statistical Physics 166 (2017), 1-23.
  joint work with Nadine Guillotin-Plantard (Lyon), Alexis Devulder (Versailles), and Françoise Pène (Brest)
   
• Small deviations of sums of correlated stationary Gaussian sequences.
  Theory of Probability and Its Applications 61 (2017), 540-568 (English).
  Teoriya Veroyatnostei i ee Primeneniya 61 (2016), 626-658 (Russian).
  joint work with Mikhail Lifshits (St. Petersburg)
   
• Upstream Polling Protocols for Flow Control in PON/xDSL Hybrid Access Networks.
  IEEE Transactions on Communications 64 (2016), 2971-2984.
  joint work of Anu Mercian, Elliot I. Gurrola, Frank Aurzada, Michael P. McGarry, Martin Reisslein.
   
• The first passage time problem over a moving boundary for asymptotically stable Lévy processes.
  Journal of Theoretical Probability 29 (2016), 737-760.
  joint work with Tanja Kramm (Darmstadt)
   
• Persistence exponent for discrete-time, time-reversible processes.
  joint work with Nadine Guillotin-Plantard (Lyon)
   
• Persistence probabilities & exponents.
  Lévy matters V, p. 183-221, Lecture Notes in Math., 2149, Springer, 2015.
  joint work with Thomas Simon (Lille)
   
• Exponential moments of first passage times and related quantities for Lévy processes.
  Mathematische Nachrichten 288 (2015), 1921-1938.
  joint work with Alexander Iksanov (Kiev) and Matthias Meiners (Darmstadt)
   
• Connection Establishment in LTE-A Networks: Justification of Poisson Process Modeling.
  IEEE Systems Journal 9 (2015), 1-12.
  joint work of Revak R. Tyagi, Frank Aurzada, Ki-Dong Lee, and Martin Reisslein.
   
• Impact of Retransmission Limit on Preamble Contention in LTE-Advanced Network.
  IEEE Systems Journal 9 (2015), 752-765.
  Revak R. Tyagi, Frank Aurzada, Ki-Dong Lee, Sang Kim, and Martin Reisslein.
   
• First passage times of Lévy processes over a one-sided moving boundary.
  Markov processes and related fields 21 (2015), 1-38.
  joint work with Tanja Kramm (Darmstadt) and Mladen Savov (Reading)
   
• DyCaPPON: Dynamic circuit and packet passive optical network.
  Optical Switching and Networking 13 (2014), 135-147.
  joint work of Xing Wei, Frank Aurzada, Michael P. McGarry, Martin Reisslein.
   
• Coexistence Analysis of H2H and M2M Traffic in FiWi Smart Grid Communications Infrastructures Based on Multi-Tier Business Models.
  IEEE Transactions on Communications 62 (2014), 3931-3942.
  joint work of Martin Lévesque, Frank Aurzada, Martin Maier, and Géza Joós.
   
• FiWi Access Networks Based on Next-Generation PON and Gigabit-Class WLAN Technologies: A Capacity and Delay Analysis.
  IEEE/ACM Transactions on Networking 22 (2014), 1176-1189.
  joint work of Frank Aurzada, Martin Lévesque, Martin Maier, and Martin Reisslein.
   
• Persistence probabilities for an integrated random walk bridge.
  Probability and Mathematical Statistics 34 (2014), 1-22.
  joint work with Steffen Dereich (Münster) and Mikhail Lifshits (St. Petersburg)
   
• Small deviations for a family of smooth Gaussian processes.
  Journal of Theoretical Probability 26 (2013), 153-168.
  joint work with Fuchang Gao (Idaho), Thomas Kühn (Leipzig), Wenbo V. Li (Delaware), and Qi-Man Shao (Hong Kong)
   
• Persistence of fractional Brownian motion with moving boundaries and applications.
  Journal of Physics A: Mathematical and Theoretical 46 (2013), 125007.
  joint work with Christoph Baumgarten (Braunschweig)
   
• EIBT: Exclusive Intervals for Bulk Transfers on EPONs.
  IEEE/OSA Journal of Lightwave Technology 31 (2013), 99-110.
  joint work of Xing Wei, Frank Aurzada, Michael P. McGarry, and Martin Reisslein.
   
• First exit of Brownian motion from a one-sided moving boundary.
  High Dimensional Probability VI: The Banff Volume. Progress in Probability 66 (2013), 215-219.
  joint work with Tanja Kramm (Braunschweig)
   
• Small time Chung type LIL for Lévy processes.
  Bernoulli 19 (2013), 115-136.
  joint work with Leif Döring (Paris VI) and Mladen Savov (Oxford)
   
• Universality of the asymptotics of the one-sided exit problem for integrated processes.
  Annales de l'Institut Henri Poincaré - Probabilités et Statistiques 49 (2013), 236-251.
  joint work with Steffen Dereich (Marburg)
   
• Wait-and-see strategies in polling models.
  Probability in the Engineering and Informational Sciences 26 (2012), 17-42.
  joint work with Sergej Beck and Michael Scheutzow (Berlin)
   
• Moments of recurrence times for Markov chains.
  Electronic Communications in Probability 16 (2011), 296-303.
  joint work with Hanna Döring, Marcel Ortgiese, and Michael Scheutzow (Berlin)
   
• On the one-sided exit problem for fractional Brownian motion.
  Electronic Communications in Probability 16 (2011), 392-404.
   
• Survival probabilities for weighted random walks.
  ALEA. Latin American Journal of Probability and Mathematical Statistics 8 (2011), 235-258.
  joint work with Christoph Baumgarten (Berlin)
   
• Capacity and Delay Analysis of Next-Generation Passive Optical Networks (NG-PONs).
  IEEE Transactions on Communications 59 (2011), 1378-1388.
  joint work of Frank Aurzada, Michael Scheutzow, Martin Reisslein, Navid Ghazisaidi, and Martin Maier.
   
• Path regularity of Gaussian processes via small deviations.
  Probability and Mathematical Statistics 31 (2011), 61-78.
   
• Intermittency and Ageing for the Symbiotic Branching Model.
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques 47 (2011), 376-394.
  joint work with Leif Döring (Berlin)
   
• Shortest Propagation Delay (SPD) First Scheduling for EPONs with Heterogeneous Propagation Delays.
  IEEE Journal on Selected Areas in Communications 28 (2010), 849-862.
  joint work of Michael P. McGarry, Martin Reisslein, Frank Aurzada, and Michael Scheutzow.
   
• Towards a Fundamental Understanding of the Stability and Delay in WDM EPONs.
  IEEE/OSA Journal of Optical Communications and Networking 2 (2010), 51-66.
  joint work of Frank Aurzada, Michael Scheutzow, Martin Reisslein, and Martin Maier.
   
• On the small deviation problem for some iterated processes.
  Electronic Journal of Probability 14 (2009), 1992-2010.
  joint work with Mikhail Lifshits (St. Petersburg)
   
• Small deviations of general Lévy processes.
  Annals of Probability 37 (2009), 2066-2092.
  joint work with Steffen Dereich (Berlin)
   
• Small deviations of stable processes and entropy of the associated random operators.
  Bernoulli 15 (2009), 1305-1334.
  joint work with Mikhail Lifshits (St. Petersburg) and Werner Linde (Jena)
   
• High resolution quantization and entropy coding of jump processes.
  Journal of Complexity 25 (2009), 163-187.
  joint work with Steffen Dereich, Michael Scheutzow, and Christian Vormoor (Berlin)
   
• The coding complexity of Lévy processes.
  Foundations of Computational Mathematics 9 (2009), 359-390.
  joint work with Steffen Dereich (Berlin)
   
• Small deviations of smooth stationary Gaussian processes.
  Theory of Probability and its Applications 53 (2009), 697-707 (English).
  Teoriya Veroyatnostei i ee Primeneniya 53 (2008), 788-798 (Russian).
  joint work with I.A. Ibragimov (St. Petersburg), M.A. Lifshits (St. Petersburg), and J.H. van Zanten (Amsterdam)
   
• Just-in-Time Scheduling for Multichannel EPONs.
  IEEE/OSA Journal of Lightwave Technology 26 (2008), 1204-1216.
  joint work of Michael P. McGarry, Martin Reisslein, Charles J. Colbourn, Martin Maier, Frank Aurzada, and Michael Scheutzow.
   
• A short note on small deviations of sequences of i.i.d. random variables with exponentially decreasing weights.
  Statistics & Probability Letters 78 (2008), 2300-2307.
   
• Small deviation probability via chaining.
  Stochastic Processes and their Applications 118 (2008), 2344-2368.
  joint work with Mikhail Lifshits (St. Petersburg)
   
• Delay Analysis of Ethernet Passive Optical Networks with Gated Service.
  OSA Journal of Optical Networking 7 (2008), 25-41.
  joint work of Frank Aurzada, Michael Scheutzow, Martin Herzog, Martin Reisslein, and Martin Maier.
   
• Small deviations for stable processes via compactness properties of the parameter set.
  Statistics & Probability Letters 78 (2008), 577-581.
   
• Metric entropy and the small deviation problem for stable processes.
  Probability and Mathematical Statistics 27 (2007), 261-274.
   
• Small ball probabilities for stable convolutions.
  ESAIM: Probability & Statistics 11 (2007), 327-343.
  joint work with Thomas Simon (Évry)
   
• On the lower tail probabilities of some random sequences in $l_p$.
  Journal of Theoretical Probability 20 (2007), 843-858.