Probability, Geometry and Integrable Systems

427 Pages·2008·7.267 MB·English

by  Mark Pinsky, Björn Birnir, Editors

Thethreemainthemesofthisbook—probabilitytheory,differentialgeometry
andthetheoryofintegrablesystems—reflectthebroadrangeofmathematical
interests of Henry McKean, to whom it is dedicated.
Writtenbyexpertsinprobability,geometry,integrablesystems,turbulence
and percolation, the seventeen papers included here demonstrate a wide va-
riety of techniques which have been developed to solve various mathematical
problems in these areas. The topics are often combined in an unusual and
interesting fashion to give solutions outside of the standard methods. The
papers contain some exciting results and offer a guide to the contemporary
literature on these subjects.

Mathematical Sciences Research Institute
Publications
55
Probability, Geometry and Integrable Systems
Mathematical Sciences Research Institute Publications
1 Freed/Uhlenbeck:Instantons and Four-Manifolds,secondedition
2 Chern(ed.):Seminar on Nonlinear Partial Differential Equations
3 Lepowsky/Mandelstam/Singer(eds.):Vertex Operators in Mathematics and Physics
4 Kac(ed.):Infinite Dimensional Groups with Applications
5 Blackadar:K-Theory for Operator Algebras,secondedition
6 Moore(ed.):Group Representations, Ergodic Theory, Operator Algebras, and
Mathematical Physics
7 Chorin/Majda(eds.):Wave Motion: Theory, Modelling, and Computation
8 Gersten(ed.):Essays in Group Theory
9 Moore/Schochet:Global Analysis on Foliated Spaces,secondedition
10–11 Drasin/Earle/Gehring/Kra/Marden(eds.):Holomorphic Functions and Moduli
12–13 Ni/Peletier/Serrin (eds.):Nonlinear Diffusion Equations and Their Equilibrium States
14 Goodman/delaHarpe/Jones:Coxeter Graphs and Towers of Algebras
15 Hochster/Huneke/Sally(eds.):Commutative Algebra
16 Ihara/Ribet/Serre(eds.):Galois Groups over Q
17 Concus/Finn/Hoffman(eds.):Geometric Analysis and Computer Graphics
18 Bryant/Chern/Gardner/Goldschmidt/Griffiths: Exterior Differential Systems
19 Alperin(ed.):Arboreal Group Theory
20 Dazord/Weinstein(eds.):Symplectic Geometry, Groupoids, and Integrable Systems
21 Moschovakis(ed.):Logic from Computer Science
22 Ratiu(ed.):The Geometry of Hamiltonian Systems
23 Baumslag/Miller(eds.):Algorithms and Classification in Combinatorial GroupTheory
24 Montgomery/Small(eds.):Noncommutative Rings
25 Akbulut/King:Topology of Real Algebraic Sets
26 Judah/Just/Woodin(eds.):Set Theory of the Continuum
27 Carlsson/Cohen/Hsiang/Jones(eds.):Algebraic Topology and Its Applications
28 Clemens/Kolla´r(eds.):Current Topics in Complex Algebraic Geometry
29 Nowakowski(ed.):Games of No Chance
30 Grove/Petersen(eds.):Comparison Geometry
31 Levy(ed.):Flavors of Geometry
32 Cecil/Chern(eds.):Tight and Taut Submanifolds
33 Axler/McCarthy/Sarason(eds.):Holomorphic Spaces
34 Ball/Milman(eds.):Convex Geometric Analysis
35 Levy(ed.):The Eightfold Way
36 Gavosto/Krantz/McCallum(eds.):Contemporary Issues in Mathematics Education
37 Schneider/Siu(eds.):Several Complex Variables
38 Billera/Bjo¨rner/Green/Simion/Stanley (eds.):New Perspectives in Geometric
Combinatorics
39 Haskell/Pillay/Steinhorn(eds.):Model Theory, Algebra, and Geometry
40 Bleher/Its(eds.):Random Matrix Models and Their Applications
41 Schneps(ed.):Galois Groups and Fundamental Groups
42 Nowakowski(ed.):More Games of No Chance
43 Montgomery/Schneider(eds.):New Directions in Hopf Algebras
44 Buhler/Stevenhagen(eds.):Algorithmic Number Theory
45 Jensen/Ledet/Yui:Generic Polynomials: Constructive Aspects of the Inverse Galois
Problem
46 Rockmore/Healy(eds.):Modern Signal Processing
47 Uhlmann(ed.):Inside Out: Inverse Problems and Applications
48 Gross/Kotiuga:Electromagnetic Theory and Computation: A Topological Approach
49 Darmon/Zhang(eds.):Heegner Points and Rankin L-Series
50 Bao/Bryant/Chern/Shen(eds.):A Sampler of Riemann–Finsler Geometry
51 Avramov/Green/Huneke/Smith/Sturmfels(eds.):Trends in Commutative Algebra
52 Goodman/Pach/Welzl(eds.):Combinatorial and Computational Geometry
53 Schoenfeld(ed.):Assessing Mathematical Proficiency
54 Hasselblatt(ed.):Dynamics, Ergodic Theory, and Geometry
55 Pinsky/Birnir(eds.):Probability, Geometry and Integrable Systems
Volumes 1–4, 6–8 and 10–27 are published by Springer-Verlag
Probability, Geometry
and Integrable Systems
For Henry McKean’s Seventy-Fifth Birthday
Edited by
Mark Pinsky
Bjo¨rn Birnir
Mark Pinsky
Department of Mathematics
Northwestern University
Evanston, IL 60208
Bj¨orn Birnir
Center for Complex and Nonlinear Science
and Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA 93106
Silvio Levy (Series Editor)
Mathematical Sciences Research Institute
17 Gauss Way, Berkeley, CA 94720
[email protected]
The Mathematical Sciences Research Institute wishes to acknowledge support by
the National Science Foundation and the Pacific Journal of Mathematicsfor the
publication of this series.
cambridge university press
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(cid:13)c Mathematical Sciences Research Institute 2008
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andtotheprovisionsofrelevantcollectivelicensingagreements,
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Probability,geometry,andintegrablesystems/editedbyMarkPinsky,Bj¨ornBirnir.
p. cm. – (Mathematical Sciences Research Institute publications ; 55)
“ForHenryMcKean’sSeventy-FifthBirthday.”Includesbibliographicalreferences
and index.
ISBN 978-0-521-89527-9 (hardback)
1.Probabilities.2.Geometry,Differential.3.Hamiltoniansystems.I.Pinsky,Mark
A., 1940- II.Birnir, Bj¨orn. III.McKean, Henry P.
QA273.P79536 2008
519.2–dc22
2007052192
ISBN 978-0-521-89527-9 hardcover
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLs
forexternalorthird-partyInternetWebsitesreferredtointhispublicationanddoesnot
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Probability,GeometryandIntegrableSystems
MSRIPublications
Volume55,2007
Contents
Preface ix
TributetoHenryMcKean xv
Directandinverseproblemsforsystemsofdifferentialequations 1
DAMIR AROV AND HARRY DYM
Turbulenceofaunidirectionalflow 29
BJORN BIRNIR
Riemann–HilbertproblemintheinversescatteringfortheCamassa–Holm
equationontheline 53
ANNE BOUTET DEMONVEL ANDDMITRY SHEPELSKY
TheRiccatimapinrandomSchro¨dingerandmatrixtheory 77
SANTIAGO CAMBRONERO, JOSE´ RAM´IREZ AND BRIAN RIDER
SLE6 andCLE6 fromcriticalpercolation 103
FEDERICO CAMIA AND CHARLES M. NEWMAN
Globaloptimization,theGaussianensembleanduniversalensembleequivalence 131
MARIUS COSTENIUC, RICHARD S. ELLIS, HUGO TOUCHETTE AND
BRUCE TURKINGTON
StochasticevolutionofinviscidBurgerfluid 167
ANA BELA CRUZEIRO AND PAUL MALLIAVIN
Aquickderivationoftheloopequationsforrandommatrices 185
N. M. ERCOLANI ANDK. D. T-R MCLAUGHLIN
SingularsolutionsforgeodesicflowsofVlasovmoments 199
J. GIBBONS, D. D. HOLM ANDC. TRONCI
Realityproblemsinsolitontheory 221
PETR G. GRINEVICH AND SERGEI P. NOVIKOV
Randomwalksandorthogonalpolynomials;somechallenges 241
F. ALBERTO GRU¨NBAUM
IntegrationofpairflowsoftheCamassa–Holmhierarchy 261
ENRIQUE LOUBET
Landensurvey 287
DANTE V. MANNA ANDVICTOR H. MOLL
vii
viii CONTENTS
Linesonabelianvarieties 321
EMMA PREVIATO
Integrablemodelsofwavesinshallowwater 345
HARVEY SEGUR
NonintersectingBrownianmotions,integrablesystemsandorthogonal
polynomials 373
PIERRE VAN MOERBEKE
HomogenizationofrandomHamilton–Jacobi–Bellmanequations 397
S. R. SRINIVASA VARADHAN
Probability,GeometryandIntegrableSystems
MSRIPublications
Volume55,2007
Preface
This volume is dedicated to Henry McKean, on the occasion of his seventy-
fifthbirthday. Hiswidespectrumofinterests withinmathematicsisreflectedin
the variety of theory and applications in these papers, discussed in the Tribute
on page xv. Here we comment briefly on the papers that make up this volume,
grouping them by topic. (The papers appear in the book alphabetically by first
author.)
Sincetheearly1970s,thesubjectofcompletelyintegrablesystemshasgrown
beyond all expectations. The discovery that the Kortweg – de Vries equation,
whichgoverns shallow-water waves,hasacomplete systemofintegrals ofmo-
tion has given rise to a search for other such evolution equations. Two of the
papers in this volume, one by Boutet deMonvelandShepelsky and the other
byLoubet, deal with the completely integrable system discovered byCamassa
andHolm. Thisequationprovidesamodeldescribingtheshallow-waterapprox-
imation in inviscid hydrodynamics. Theunknown function u.x;t/refers tothe
horizontal fluid velocity along the x-direction at time t. The first authors show
that the solution of the CH equation in the case of no breaking waves can be
expressedinparametricformintermsofthesolutionofanassociatedRiemann–
Hilbertproblem. Thisanalysisallowsonetoconcludethateachsolutionwithin
this class develops asymptotically into a train of solitons.
Loubet provides a technical tour de force, in extending previous results of
McKeanontheCamassa–Holmequation. Morespecifically,hegivesanexplicit
formula for the velocity profile in terms of its initial value, when the dynamics
aredefinedbyaHamiltonian whichisthesumofthesquaresofthereciprocals
ofapairofeigenvaluesofanassociatedacousticequation. Theproofdependson
theanalysisofasimplersystem,whoseHamiltonianisdefinedbythereciprocal
of a single eigenvalue of the acoustic equation. This tool can also solve more
complex dynamics, associated to several eigenvalues, which eventually leads
to a new proof of McKean’s formula for the Fredholm determinant. The paper
concludeswithanasymptoticanalysis(inbothpastandfuturedirections),which
allows partial confirmation of statements about soliton genesis and interaction
which wereraised inanearlier CPAM paper.
ix
x PREFACE
Meanwhile we have a contribution from Gibbons, Holm and Tronci of a
more geometric nature. This deals with the Vlaslov equation, which describes
the evolution of the single-particle probability density in the evolution of N
pointparticles. Specifically,theystudytheevolutionofthep-thmomentswhen
the dynamics is governed by a quadratic Hamiltonian. The resulting motion
takes place on the manifold of symplectomorphisms, which are smooth invert-
iblemapsactingonthephasespace. Thesingularsolutionsturnouttobeclosely
related to integrable systems governing shallow-water wave theory. In fact,
when these equations are “closed” at level p, one retrieves the peaked solitons
of the integrable Camassa–Holm equation forshallow-water waves!
Segur’s paper provides an excellent overview of the development of our
understanding of integrable partial differential equations, from the Boussinesq
equation (1871) to the Camassa–Holm equation (1993) and their relations to
shallow-waterwavetheory. Inphysicalterms,anintegrablesystemisequivalent
to the existence of action-angle variables, where the action variables are the
integrals ofmotionandtheanglevariables evolveaccordingtosimpleordinary
differentialequations. SinceeachofthesePDEsalsodescribeswavesinshallow
water,itisnaturaltoaskthequestion: Doestheextramathematicalstructureof
complete integrability provide useful information about the behavior of actual
physicalwavesinshallowwater? Thebodyofthepapertakesupthisquestionin
detail with many illustrations of real cases, including the tsunami of December
26,2004. A video link is provided, forfurther documentation.
Previato’s paper contains a lucid account of the use of theta functions to
characterize lines in abelian varieties. Strictly speaking, “line” is short for lin-
ear flow, since an abelian variety cannot properly contain a (projective) line.
More than twenty years ago Barsotti had proved that, on any abelian variety,
thereexistsadirection suchthatthederivatives ofsufficiently highorderofthe
logarithimofthethetafunctiongeneratesthefunctionfieldoftheabelianvariety.
Thepurposeofthepresentpaperistousetheresultingdifferential equationsto
characterizethetafunctions,ageneralizationoftheKPequations,introducedby
KadomtsevandPetviashviliin1971,aswellastostudyspectraofcommutative
rings of partial differential operators.
ArovandDymsummarizetheirrecentworkoninverseproblemsformatrix-
valuedordinarydifferentialequations. Thisisrelatedtothenotionofreproduc-
ing kernel Hilbert spaces and the theory of J-inner matrix functions. A time-
independentSchro¨dingerequationiswrittenasasystemoffirst-orderequations,
which permits application of thebasic results.
CruzeiroandMalliavinstudyafirst-orderBurgersequationinthecontextof
flowsonthegroupofdiffeomorphismsofthecircle,aninfinite-dimensionalRie-
2
mannianmanifold. TheL normonthecircledefinestheRiemannianmetric,so