Table Of ContentUniversitext
Marta Lewicka
A Course on
Tug-of-War
Games with
Random Noise
Introduction and Basic Constructions
Universitext
Universitext
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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
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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:
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licencetoSpringerNatureSwitzerlandAG2020
M.Lewicka,ACourseonTug-of-WarGameswithRandomNoise,Universitext,
https://doi.org/10.1007/978-3-030-46209-3_1
