Seminaire de Probabilites XLIX

544 Pages·2018·3.376 MB·English

by  Donati-Martin C (ed.)

Lecture Notes in Mathematics 2215
Séminaire de Probabilités
Catherine Donati-Martin 
Antoine Lejay · Alain Rouault Editors
Séminaire de
Probabilités
XLIX
Lecture Notes in Mathematics 2215
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Catherine Donati-Martin (cid:129) Antoine Lejay (cid:129)
Alain Rouault
Editors
Séminaire de Probabilités
XLIX
123
Editors
CatherineDonati-Martin AntoineLejay
LaboratoiredeMathématiquesdeVersailles InstitutElieCartandeLorraine
UniversitéVersaillesSaint-Quentin Vandoeuvre-les-Nancy,France
Versailles,France
AlainRouault
LaboratoiredeMathématiquesdeVersailles
UniversitéVersaillesSaint-Quentin
Versailles,France
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-92419-9 ISBN978-3-319-92420-5 (eBook)
https://doi.org/10.1007/978-3-319-92420-5
LibraryofCongressControlNumber:2018950488
MathematicsSubjectClassification(2010):60G,60J,60K
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Preface
In this 49th volume we continue to offer a good sample of the main streams of
currentresearch on probabilityand stochastic processes, in particular those active
in France. All the contributions come from spontaneous submissions and their
diversityillustratesthegoodhealthofthisbranchofmathematics.
Since the publication of the 48th volume, we have received two sad pieces of
news:
Jacques Neveu, former professor at the Université Pierre et Marie Curie and
École Polytechnique, passed away on May 15, 2016. His influence on the strong
development of probability in France was huge. For details and testimonies, we
referto“HommagesàJacquesNeveu”,asupplementtoMatapli112,52pp.,2017,
seethewebsitehttp://smai.emath.fr/IMG/pdf/Matapli_J_Neveu.pdf.
RonGetoor,professorattheUniversityofCalifornia,SanDiego,passedawayon
October28,2017.Hewasaleaderinthegrowthofprobabilitytheory.Fordetails,
seethewebsitehttps://www.math.ucsd.edu/memorials/ronald-getoor/.
BothofthempublishedexcellentbooksandcontributionstoTheSéminaire.We
would like to remind the reader that the website of the Séminaire is http://sites.
mathdoc.fr/SemProba/ and that all the articles of the Séminaire from Volume I
(1967)toVolumeXXXVI(2002)arefreelyaccessiblefromthewebsitehttp://www.
numdam.org/actas/SPS.
WethanktheCelluleMathDocforhostingthesearticleswithintheNUMDAM
project.
Versailles,France CatherineDonati-Martin
Vandoeuvre-lès-Nancy,France AntoineLejay
Versailles,France AlainRouault
v
Contents
1 Ornstein-UhlenbeckPinball and the Poincaré Inequality
inaPuncturedDomain .................................................... 1
Emmnuel Boissard, Patrick Cattiaux, Arnaud Guillin,
andLaurentMiclo
2 AProbabilisticLookatConservativeGrowth-Fragmentation
Equations.................................................................... 57
FlorianBouguet
3 IteratedProportionalFittingProcedureandInfiniteProducts
ofStochasticMatrices...................................................... 75
J.BrossardandC.Leuridan
4 Limiting Eigenvectors of Outliers for Spiked
Information-Plus-NoiseTypeMatrices .................................. 119
MireilleCapitaine
5 CriteriaforExponentialConvergencetoQuasi-Stationary
DistributionsandApplicationstoMulti-DimensionalDiffusions..... 165
Nicolas Champagnat,Koléhè AbdoulayeCoulibaly-Pasquier,
andDenisVillemonais
6 Bismut-Elworthy-LiFormulaeforBesselProcesses.................... 183
HenriEladAltman
7 LargeDeviationsforInfectiousDiseasesModels ....................... 221
PeterKratzandEtiennePardoux
8 TheGirsanovTheoremWithout(SoMuch)StochasticAnalysis ..... 329
AntoineLejay
9 OnDriftingBrownianMotionMadePeriodic .......................... 363
PaulMcGill
vii
viii Contents
10 OntheMarkovianSimilarity ............................................. 375
LaurentMiclo
11 SharpRatefortheDualQuantizationProblem......................... 405
GillesPagèsandBenediktWilbertz
12 Cramér’sTheoreminBanachSpacesRevisited ........................ 455
PierrePetit
13 OnMartingaleChaoses.................................................... 475
B.Rajeev
14 ExplicitLawsfortheRecordsofthePerturbedRandomWalk
onZ .......................................................................... 495
LaurentSerlet
15 A Potential Theoretic Approach to Tanaka Formula
forAsymmetricLévyProcesses........................................... 521
HiroshiTsukada
Chapter 1
Ornstein-Uhlenbeck Pinball
and the Poincaré Inequality
in a Punctured Domain
EmmnuelBoissard,PatrickCattiaux,ArnaudGuillin,andLaurentMiclo
Abstract In this paper we study the Poincaré constant for the Gaussian measure
restrictedtoD =Rd −BwhereBisthedisjointunionofboundedopensets.We
willmainlylookatthecasewheretheobstaclesareEuclideanballsB(x ,r )with
i i
radiir ,orhypercubeswithverticesoflength2r ,andd ≥ 2.Thiswillexplainthe
i i
asymptoticbehaviorofad-dimensionalOrnstein-Uhlenbeckprocessinthepresence
ofobstacleswithelasticnormalreflections(theOrnstein-Uhlenbeckpinball).
Keywords Poincaréinequalities · Lyapunovfunctions · Hittingtimes ·
Obstacles
MSC2010 26D10,39B62,47D07,60G10,60J60
1.1 Introduction
Inorderto understandthe goalofthepresentpaperletusstartwith a wellknown
question: how many non overlapping unit discs can be placed in a large square
S? This problem of discs packing has a very long history including the following
otherquestion:isitpossibletoperformanalgorithmyieldingtoaperfectlyrandom
configurationofN suchdiscsatasufficientlyquickrate(exponentialforinstance)?
ThisisoneoftheoriginoftheMetropolisalgorithmsasrefereedin[16].
The meaning of perfectly random is the following: the configuration space for
the modelis SN, describingthe locationofthe N centersofthe N discsB(x ,1),
i
E.Boissard·P.Cattiaux((cid:2))·L.Miclo
InstitutdeMathématiquesdeToulouse,CNRSUMR5219,UniversitéPaulSabatier,Toulouse
Cedex,France
e-mail:[email protected];[email protected]
A.Guillin
LaboratoiredeMathématiques,CNRSUMR6620,UniversitéBlaisePascal,Aubière,France
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2018 1
C.Donati-Martinetal.(eds.),SéminairedeProbabilitésXLIX,
LectureNotesinMathematics2215,https://doi.org/10.1007/978-3-319-92420-5_1
2 E.Boissardetal.
but under the constraints d(x ,∂S) ≥ 1 and for all i (cid:3)= j, |x − x | ≥ 2. The
i i j
remaining domain D is quite complicated, and randomness is described by the
uniformmeasureonD.
Theanswertothesecondquestionispositive,essentiallythankstocompactness,
buttheexponentintheexponentialrateofconvergenceisstronglyconnectedwith
thePoincaréconstantfortheuniformmeasureonD whichis,atthepresentstage,
fartobeknown(theonlyknownupperboundsaredisastrous).
One can of course ask the same questions replacing the square by the whole
Euclideanspace,andtheuniformmeasurebysomenaturalprobabilitymeasure,for
instancetheGaussianone.ButthistimeeventhefinitenessofthePoincaréconstant
isnomoreclear.Averypartialstudy(N =2,3)ofthisproblemisdonein[15].
In all cases, the probability measure under study, and supported by the com-
plicated state space D is actually an invariant(even reversible)measure for some
Markovian dynamics, one can study by itself, and which furnishes a possible
algorithm.TheboundaryofDbecomesareflectingboundaryforthedynamics.
In this paper we intend to study the asymptotic behavior of a d-dimensional
Ornstein-Uhlenbeck process in the presence of bounded obstacles with elastic
normal reflections (looking like a random pinball). The choice of an Ornstein-
Uhlenbeck(henceofaninvariantmeasureofGaussiantype)ismadeforsimplicity
as it capturesalreadyall the new difficultiesof this setting, buta generalgradient
driftdiffusionprocess(satisfyinganordinaryPoincaréinequality)couldbeconsid-
ered.
Of course for the packing problem in the whole space the obstacles are not
bounded,but it seems interesting to look first at the present setting. Our modelis
alsomotivatedbyothersconsiderationsweshallgivelater.
All over the paper we assume that d ≥ 2. We shall mainly consider the
∞
case where the obstacles are non overlapping Euclidean balls or smoothed l
balls (hence smoothed hypercubes) of radius ri and centers (xi)1≤i≤N≤+∞, as
overlappingobstaclescouldproducedisconnecteddomainsandthusnonuniqueness
of invariant measures (as well as no Poincaré inequality). We shall also look at
differentformsofobstacleswhenitcanenlightenthediscussion.
To be more precise, consider for 1 ≤ N ≤ +∞, X = (xi)1≤i≤N≤+∞ a
locally finite collection of points, and (ri)1≤i≤N≤+∞ a collection of non negative
realnumbers,satisfying
|x −x |>r +r fori (cid:3)=j. (1.1)
i j i j
TheOrnstein-Uhlenbeckpinballwillbegivenbythefollowingstochasticdifferen-
tialsystemwithreflection
(cid:2) (cid:3)
dXt =(cid:4)dWt − λXtdt + i(Xt −xi)dLit, (1.2)
Lit = 0t 1|Xs−xi|=ridLis.