A Course on Tug-of-War Games with Random Noise: Introduction and Basic Constructions PDF
Preview A Course on Tug-of-War Games with Random Noise: Introduction and Basic Constructions
Universitext Marta Lewicka A Course on Tug-of-War Games with Random Noise Introduction and Basic Constructions Universitext Universitext SeriesEditors SheldonAxler SanFranciscoStateUniversity CarlesCasacuberta UniversitatdeBarcelona JohnGreenlees UniversityofWarwick AngusMacIntyre QueenMaryUniversityofLondon KennethRibet UniversityofCalifornia,Berkeley ClaudeSabbah ÉcolePolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau EndreSüli UniversityofOxford WojborA.Woyczyn´ski CaseWesternReserveUniversity Universitext is a series of textbooks that presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,intoverypolishedtexts. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/223 Marta Lewicka A Course on Tug-of-War Games with Random Noise Introduction and Basic Constructions MartaLewicka DepartmentofMathematics UniversityofPittsburgh Pittsburgh,PA,USA ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-030-46208-6 ISBN978-3-030-46209-3 (eBook) https://doi.org/10.1007/978-3-030-46209-3 MathematicsSubjectClassification(2020):91A15,91A24,31C45,35B30,35G30,35J70 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicencetoSpringerNatureSwitzerland AG2020 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The goal of these Course Notes is to present a systematic overview of the basic constructions and results pertaining to the recently emerged field of Tug-of- War games, as seen from an analyst’s perspective. To a large extent, this book represents the author’s own study itinerary, aiming at precision and completeness ofaclassroomtextinanupperundergraduate-tograduate-levelcourse. This book was originally planned as a joint project between Marta Lewicka (University of Pittsburgh) and Yuval Peres (then Microsoft Research). Due to an unforeseenturnofevents,neitherthecollaborationnortheexecutionoftheproject intheirpriorlyconceivedformscouldhavebeenpursued. The author wishes to dedicate this book to all women in mathematics, with admiration and encouragement. The publishing profit will be donated to the AssociationforWomeninMathematics. Pittsburgh,PA,USA MartaLewicka October2019 v Contents 1 Introduction .................................................................. 1 2 TheLinearCase:RandomWalkandHarmonicFunctions............. 9 2.1 TheLaplaceEquationandHarmonicFunctions...................... 10 2.2 TheBallWalk .......................................................... 11 2.3 TheBallWalkandHarmonicFunctions .............................. 17 2.4 ConvergenceattheBoundaryandWalk-Regularity.................. 19 2.5 ASufficientConditionforWalk-Regularity .......................... 23 2.6* TheBallWalkValuesandPerronSolutions .......................... 27 2.7* TheBallWalkandBrownianTrajectories ............................ 29 2.8 BibliographicalNotes.................................................. 34 3 Tug-of-WarwithNoise:Casep ∈[2,∞).................................. 37 3.1 Thep-HarmonicFunctionsandthep-Laplacian..................... 38 3.2 TheAveragingPrinciples .............................................. 40 3.3 TheFirstAveragingPrinciple.......................................... 47 3.4 Tug-of-WarwithNoise:ABasicConstruction ....................... 51 3.5 TheFirstAveragingastheDynamicProgrammingPrinciple........ 57 3.6* Casep =2andBrownianTrajectories ............................... 60 3.7* EquivalenceofRegularityConditions................................. 65 3.8 BibliographicalNotes.................................................. 68 4 BoundaryAwareTug-of-WarwithNoise:Casep ∈(2,∞)............. 71 4.1 TheSecondAveragingPrinciple....................................... 72 4.2 TheBasicConvergenceTheorem...................................... 77 4.3 PlayingBoundaryAwareTug-of-WarwithNoise.................... 82 4.4 AProbabilisticProofoftheBasicConvergenceTheorem ........... 88 4.5* TheBoundaryAwareProcessatp =2andBrownian Trajectories ............................................................. 93 4.6 BibliographicalNotes.................................................. 99 vii viii Contents 5 Game-RegularityandConvergence:Casep ∈(2,∞)................... 101 5.1 Convergencetop-HarmonicFunctions............................... 102 5.2 Game-RegularityandConvergence.................................... 109 5.3 ConcatenatingStrategies............................................... 113 5.4 TheAnnulusWalkEstimate ........................................... 117 5.5 SufficientConditionsforGame-Regularity:Exterior ConeProperty .......................................................... 120 5.6 SufficientConditionsforGame-Regularity:p >N.................. 122 5.7 BibliographicalNotes.................................................. 134 6 MixedTug-of-WarwithNoise:Casep ∈(1,∞) ......................... 135 6.1 TheThirdAveragingPrinciple......................................... 136 6.2 The Dynamic Programming Principle and the Basic ConvergenceTheorem ................................................. 142 6.3 MixedTug-of-WarwithNoise......................................... 146 6.4 Sufficient Conditions for Game-Regularity: Exterior CorkscrewCondition................................................... 152 6.5 Sufficient Conditions for Game-Regularity: Simply ConnectednessinDimensionN =2.................................. 156 6.6 BibliographicalNotes.................................................. 161 A BackgroundinProbability.................................................. 163 A.1 ProbabilityandMeasurableSpaces.................................... 163 A.2 RandomVariablesandExpectation.................................... 165 A.3 ProductMeasures....................................................... 168 A.4 ConditionalExpectation................................................ 172 A.5 Independence........................................................... 174 A.6 MartingalesandStoppingTimes ...................................... 176 A.7 ConvergenceofMartingales ........................................... 180 B BackgroundinBrownianMotion.......................................... 185 B.1 DefinitionandConstructionofBrownianMotion .................... 185 B.2 TheWienerMeasureandUniquenessofBrownianMotion.......... 192 B.3 TheMarkovProperties................................................. 194 B.4 BrownianMotionandHarmonicExtensions ......................... 199 C BackgroundinPDEs ........................................................ 203 C.1 LebesgueLp SpacesandSobolevW1,p Spaces...................... 203 C.2 SemicontinuousFunctions............................................. 209 C.3 HarmonicFunctions.................................................... 210 C.4 Thep-LaplacianandItsVariationalFormulation..................... 214 C.5 WeakSolutionstothep-Laplacian.................................... 217 Contents ix C.6 PotentialTheoryandp-HarmonicFunctions ......................... 221 C.7 BoundaryContinuityofWeakSolutionsto(cid:2) u=0................ 224 p C.8 ViscositySolutionsto(cid:2) u=0....................................... 229 p D SolutionstoSelectedExercises ............................................. 231 References......................................................................... 249 Index............................................................................... 253 Chapter 1 Introduction The goal of these Course Notes is to present a systematic overview of the basic constructions pertaining to the recently emerged field of Tug-of-War games with randomnoise,asseenfromananalyst’sperspective. TheLinearMotivation Theprototypicalellipticequation,arisingubiquitouslyin Analysis,FunctionTheoryandMathematicalPhysics,istheLaplaceequation: (cid:2) (cid:3) . (cid:2)u=div ∇u =0. Laplace’s equation is linear and its solutions u : D → R, defined on a domain D ⊂ RN,arecalledharmonicfunctions.Theyarepreciselythecriticalpoints(i.e., solutionstotheEuler–Lagrangeequation)ofthequadraticpotentialenergy: ˆ . I (u)= |∇u(x)|2dx. 2 D TheideabehindtheclassicalinterplayofthelinearPotentialTheoryandProbability isthatharmonicfunctionsandrandomwalksshareacommonaveragingproperty. Wenowbrieflyrecallthisrelation. Indeed,ontheonehand,foranyu∈C2(RN)thereholdstheexpansion: (cid:3)2 u(y)dy =u(x)+ (cid:2)u(x)+o((cid:3)2) as (cid:3) →0+. 2(N +2) B(cid:3)(x) ffl ´ . Recall that u = 1 u denotes the average of u on a set B. The displayed B |B| B formulafollowsbywritingthequadraticTaylorexpansionofuatx: ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusive 1 licencetoSpringerNatureSwitzerlandAG2020 M.Lewicka,ACourseonTug-of-WarGameswithRandomNoise,Universitext, https://doi.org/10.1007/978-3-030-46209-3_1
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