Probability and Stochastic Processes Indian Statistical Institute, Bangalore August 13-17, 2010
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Collision Times of Random Walks and Applications to the Brownian
Collision times of random walks and applications to the Brownian web David Coupier1, Kumarjit Saha2, Anish Sarkar3, and Viet Chi Tran4 1Univ. de Valenciennes, CNRS 2Ashoka University 3Indian Statistical Institute, Delhi 4Univ. Lille, CNRS February 12, 2019 Abstract Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their conver- gence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length t(> 0), all starting within a segment of length ε, is of small order of ε. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first col- lision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([22], [14], [11]). -
New Beginnings
NEW BEGINNINGS A three-year report of Eklavya Foundation 2001-2004 NEW BEGINNINGS A three vi'ar report of the Ikb\\a I ouruiatinn fJQOl Jun/j i We gratffulli a. k now ledpc tor Design. IOVLM illu-itMtion & le<if motif: Urijii Moh.mly. IDC. NT F'OVM^I, Murnbai Sketches Mrm.r.wmi 1 ^td Ravnuird & AntdM HjzanK.i Murnhni ic This honor, uy piojoU Inciu'.ti i.il D ,icjn C^ntif, MF P'j\\-ii. Mun.h.ii Eklavya Foundation H(Hji'.U>u>rJ Vu itMy No S l *n I ) H U--H .-/> 1 lh.' 1 ' 1 -- Mi'qiM(M(KlUtfic" H .' i. .VKM!. W.M U..lhi , O'uj ' ' ".." i " ' AM (l(.jt|.|ir In | U.'.-v.i i" 't!|i l-i, mi*, i.j, ,M.-t ,"H.'I.. :i ] J,)I)M 1- * ' 1 >' ' l I ; v'lil- '-! i i- ' .. ii i'. .i J /()(.)/ [ '! .'11" ',-,* ? ;i' ! \ i i i I)' .'. )! _>f [ ^h Jciiuuirv <?Ol)S '.'Jon uj l ontnhutoiy put i <-, sn Published In Eklavya L / Hir,-1',^ Arer.u olony Phone 0755 246 3380, 246 4824 - Fax: 0755 246 1 703 Email eklavyampfj'mantrdfteenet corn Printed at ' Bhanclan Otis*.". Pnnti-rs Arfj v.n!c>nv Bliou.u P ( iont- /-it; i Contents The dismantling oKwiMon People's voices Report of cK-tiviLies(;OOl Foreword In this report we share with you a truly momentous leg of our long journey Two events impacted significantly on the structure and work of Eklavya. One was the closure of our collaborative middle school educational programmes by the Madhya Pradesh government in July 2002 The second was the bifurcation of Eklavya into two autonomous units - a 'school education and publication' group and a'rural development'group.The latter was registered as a new society on August 1 1, 2003 as the Samavesh Society for Development and Governance. -
ICTS POSTER Outside Bangalore
T A T A I N S T I T U T E O F F U N D A M E N T A L R E S E A R C H A HOMI BHABHA BIRTH CENTENARY & ICTS INAUGURAL EVENT International Centre Theoretical Sciences science without bo28 Decemberun 2009d29 -a 31 Decemberri e2009s Satish Dhawan Auditorium Faculty Hall Indian Institute of Science, Bangalore. www.icts.res.in/program/icts-ie INVITED SPEAKERS / PANELISTS INCLUDE FOUNDATION STONE CEREMONY Siva Athreya ISI, Bangalore OF ICTS CAMPUS Naama Barkai Weizmann Institute The foundation stone will be unveiled by Manjul Bhargava Princeton University Prof. C N R Rao, FRS 4:00 pm, 28 December 2009 Édouard Brézin École Normale Supérieure Amol Dighe TIFR Michael Green DAMTP, Cambridge Chandrashekhar Khare UCLA Yamuna Krishnan NCBS-TIFR Lyman Page Princeton University Jaikumar Radhakrishnan TIFR C. S. Rajan TIFR Sriram Ramaswamy IISc G. Rangarajan IISc C. N. R. Rao JNCASR Subir Sachdev Harvard University K. Sandeep CAM-TIFR Sriram Shastry UC Santa Cruz PUBLIC LECTURES Ashoke Sen HRI J. N. Tata Auditorium, IISc (FREE AND OPEN TO ALL) Anirvan Sengupta Rutgers University K. R. Sreenivasan Abdus Salam ICTP Michael Atiyah University of Edinburgh Andrew Strominger Harvard University Truth and Beauty in Mathematics and Physics 5:30 pm, 27 December 2009 Raman Sundrum Johns Hopkins University Ajay Sood IISc David Gross KITP, Santa Barbara The Role of Theory in Science Tarun Souradeep IUCAA 5:30 pm, 28 December 2009 Eitan Tadmor University of Maryland Albert Libchaber Rockefeller University Sandip Trivedi TIFR The Origin of Life: from Geophysics to Biology? Mukund Thattai NCBS-TIFR 5:30 pm, 30 December 2009 S. -
Abramo Hefez
We certify that Abramo Hefez --- Universidade Federal Fluminense participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Singularities“Singularities of analytic plane curvescurves”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Adimurthi --- Tata Institute of Fundamental Research participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Hardy“Hardy Soblev Rellich InequalitiesInequalities”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Antar BandyopadBandyopadhyayhyay --- Indian Statistical Institute, Delhi Centre participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Annealed“Annealed and Quenched IP for Random Walk in Dynamic Markovian EnvironmentEnvironment”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Antonio Roberto da Silva --- Universidade Federal do Rio de Janeiro participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Modified“Modified symplectic structures in cotangent bundlesbundles of Lie groupsgroups”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Arnaldo Leite Pinto Garcia --- Instituto Nacional de Matemática Pura e Aplicada participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““On“On curves and towers over finite fieldsfields”””” Rio de Janeiro, July 31, 2008. -
Academic Review
ACADEMIC REVIEW for the period April 2006 { March 2011 Department of Mathematics Indian Institute of Science Bangalore May 13, 2011 Prepared by Basudeb Datta Contents 1 About the Department 1 1.1 Institute and Department background ................... 1 1.2 Scientific Staff in the Department ...................... 2 1.2.1 Faculty .................................... 2 1.2.2 Honorary Professors ............................ 2 1.2.3 Affiliated Scientists ............................. 3 1.2.4 Visiting Professors .............................. 3 1.2.5 MO Cell Faculty .............................. 3 1.2.6 Post Doctoral Fellows/Research Associates ................ 4 1.2.7 Students ................................... 5 1.2.8 Visiting Students .............................. 10 2 Thrust area during DSA-IV under SAP 11 2.1 Faculty involved (in the identified thrust area) . 11 3 Major achievements 11 3.1 New Ph.D. Programme ............................ 11 3.2 Teaching ...................................... 11 3.3 Research ...................................... 17 3.3.1 Research Highlights in Thrust area ..................... 17 3.3.2 Research Highlights in other areas ..................... 37 3.3.3 Publications ................................. 44 3.4 Human resource training ........................... 74 3.4.1 The IISc Mathematics Initiative (IMI) ................... 74 3.4.2 Indo-French Cyber University Project .................... 75 3.4.3 Mathematical Olympiad ........................... 75 3.4.4 Olympiad training camps .......................... 76 3.4.5 Visiting summer students .......................... 76 4 Infrastructure developed 77 5 Knowledge disseminated (in the thrust area identified) 77 5.1 Seminar Series for undergraduate students . 77 5.2 Outreach activities for students and teachers . 78 6 Resource generation 78 7 Others 82 7.1 Awards/Distinctions .............................. 82 7.1.1 Awards .................................... 82 7.1.2 Fellowships and Honours .......................... 82 7.1.3 Awards/Distinctions conferred on students . -
The Brownian Web
The Annals of Probability 2008, Vol. 36, No. 3, 1153–1208 DOI: 10.1214/07-AOP357 c Institute of Mathematical Statistics, 2008 THE BROWNIAN NET By Rongfeng Sun and Jan M. Swart1 TU Berlin and UTIA´ Prague The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties. Contents 1. Introduction and main results....................... .....................1154 1.1. Arrow configurations and branching-coalescing random walks.......1154 1.2. Topology and convergence......................... .................1155 1.3. The Brownian web................................. ................1158 1.4. Characterization of the Brownian net using hopping . .............1159 1.5. The left-right Brownian web....................... .................1160 1.6. Characterization of the Brownian net using meshes . .............1161 1.7. The dual Brownian web............................. ...............1162 1.8. Dual characterization of the Brownian net........... ...............1164 1.9. The branching-coalescing point set ................ ................. 1164 1.10. The backbone................................... ...................1166 1.11. Discussion, applications and open problems . .................1167 -
The Brownian Web
The Brownian Web Linus Mattauch University of Oxford, Honour School of Mathematics and Philosophy, Part C Dissertation - March 2008 1 Acknowledgements I am extremely grateful to my supervisors, Alison Etheridge and Pierre Tarr`es,for guiding me through the challenging material on which this dissertation is based and for their patient explanations of mathematical details. I wish to thank R. Arratia for making a seminal unpublished manuscript [1] available to me. I am very grateful to R. Sun for helpful comments (in particular regarding [30, 31]) and explanation of a detail in [12]. I am, moreover, very grateful to Arend Janssen for reading a draft of this disser- tation and to my parents, Angela and Hans Mattauch, for proof-reading and encour- agement. Special mention goes to David Yadin for constant support and to Glenys Luke for giving wise advice whenever needed. Finally I wish to thank Robert Goudie and Tsubasa Itani for some help with LATEX and everyone else who contributed to a fantastic time as an undergraduate at St. Hugh's College. Linus Mattauch March 2008 Note concerning the proposal I wish to note that my study of the Brownian web turned out to be somewhat less connected to the theory of interacting particle systems than I anticipated at the time of submitting the proposal. So the connection between the voter model and coalescing random walks is included in this dissertation (as Appendix A), but the focus of the present work is exclusively on the Brownian web itself. An alternative for the end of the project was stated in the proposal: I decided to include the topic of the \Poisson tree" (not that of self-interacting random walks) into this dissertation and also tailored my account of the Brownian web accordingly. -
The Brownian Fan
The Brownian fan April 9, 2014 Martin Hairer1 and Jonathan Weare2 1 Mathematics Department, the University of Warwick 2 Statistics Department and the James Frank Institute, the University of Chicago Email: [email protected], Email: [email protected] Abstract We provide a mathematical study of the modified Diffusion Monte Carlo (DMC) algorithm introduced in the companion article [HW14]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman- Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the “na¨ıve” generalisation of the DMC algorithm which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the “Brownian fan”, is a very natural new mathematical object of independent interest. Keywords: Diffusion Monte Carlo, quantum Monte Carlo, rare event simulation, sequential Monte Carlo, particle filtering, Brownian fan, branching process Contents 1 Introduction 2 1.1 Notations . .5 2 The continuous-time limit and the Brownian fan 5 2.1 Heuristic derivation of the continuous-time limit . .6 2.2 Some properties of the limiting process . -
Mathematics of the Gateway Arch Page 220
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice. -
Annual Report
THE INSTITUTE OF MATHEMATICAL SCIENCES C. I. T. Campus, Taramani, Chennai - 600 113. ANNUAL REPORT Aug 2001 - Jul 2002 Telegram: MATSCIENCE Fax: +91-44-254 1586 Telephone: +91-44-254 2398, 254 1856, 254 2588, 254 1049, 254 2050 e-mail: offi[email protected] ii Foreword I am pleased to present the progress made by the Institute during 2001-2002 in its many sub-disciplines and note the distinctive achievements of the members of the Institute. The construction of additional office space and an auditorium was completed and the new office space has now been occupied. The Auditorium, which has a seating capacity of 190, was named the Ramanujan Auditorium and was inaugurated on June 22nd by Dr. Anil Kakodkar. On behalf of the Institute, I convey my sincere thanks to Dr. R. Chidambaram (the previous Chairman of AEC), Dr. Anil Kakodkar (the present Chairman), and Mrs. Sudha Bhave (Joint Secretary, DAE), for the keen interest shown by them in the progress of this building project and the help and support extended to us at every stage. I am pleased to mention that 2001-2002 was an academically productive year and many interesting events took place. The B. M. Birla Award was given to Dr. Kapil Paranjape for Mathematical Science for the year 1999, and to Dr. Sudeshna Sinha for Physical Sciences for the year 2001. Dr. Kapil Paranjape was awarded a DST-Swarnajayanthi Project on Cycles, Arithmetic and Cryptography. Prof. K. Srinivasa Rao was awarded the Tamil Nadu Scientists Award for the year 2000 by the Tamil Nadu State Council for Science and Technology. -
Recent Progress in Coalescent Theory
SOCIEDADEBRASILEIRADEMATEMÁTICA ENSAIOS MATEMATICOS´ 200X, Volume XX, X–XX Recent Progress in Coalescent Theory Nathana¨el Berestycki Abstract. Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathe- matics of coalescent processes and their applications to theoretical population genetics and other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks. arXiv:0909.3985v1 [math.PR] 22 Sep 2009 2000 Mathematics Subject Classification: 60J25, 60K35, 60J80. Introduction The probabilistic theory of coalescence, which is the primary subject of these notes, has expanded at a quick pace over the last decade or so. I can think of three factors which have essentially contributed to this growth. On the one hand, there has been a rising demand from population geneticists to develop and analyse models which incorporate more realistic features than what Kingman’s coalescent allows for. Simultaneously, the field has matured enough that a wide range of techniques from modern probability theory may be success- fully applied to these questions. These tools include for instance martingale methods, renormalization and random walk arguments, combinatorial embeddings, sample path analysis of Brownian motion and L´evy processes, and, last but not least, continuum random trees and measure-valued processes. Finally, coalescent processes arise in a natural way from spin glass models of statistical physics. The identification of the Bolthausen-Sznitman coalescent as a universal scaling limit in those models, and the connection made by Brunet and Derrida to models of population genetics, is a very exciting re- cent development. -
Lecture Notes in Mathematics 1840 Editors: J.--M.Morel, Cachan F
Lecture Notes in Mathematics 1840 Editors: J.--M.Morel, Cachan F. Takens, Groningen B. Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Boris Tsirelson Wendelin Werner Lectures on Probability Theory and Statistics Ecole d’EtedeProbabilit´ es´ de Saint-Flour XXXII - 2002 Editor: Jean Picard 13 Authors Editor Boris Tsirelson Jean Picard School of Mathematics Laboratoire de Mathematiques´ Appliquees´ TelAvivUniversity UMR CNRS 6620 Tel Av iv 69978 Universite´ Blaise Pascal Clermont-Ferrand Israel 63177 Aubiere` Cedex, France e-mail: [email protected] e-mail: [email protected] Wendelin Werner Laboratoire de Mathematiques´ Universite´ Paris-Sud Batˆ 425, 91405 Orsay Cedex France e-mail: [email protected] Cover picture: Blaise Pascal (1623-1662) Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2001): 60-01, 60Gxx, 60J65, 60K35, 82B20, 82b27, 82B41 ISSN 0075-8434 Lecture Notes in Mathematics ISSN 0721-5363 Ecole d’Ete´ des Probabilites´ de St. Flour ISBN 3-540-21316-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.