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Convexities and Optimal Transport Problems on the Wiener Space Vincent Nolot

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Convexities and Optimal Transport Problems on the Wiener Space Vincent Nolot Convexities and optimal transport problems on the Wiener space Vincent Nolot To cite this version: Vincent Nolot. Convexities and optimal transport problems on the Wiener space. General Mathe- matics [math.GM]. Université de Bourgogne, 2013. English. NNT : 2013DIJOS016. tel-00932092 HAL Id: tel-00932092 https://tel.archives-ouvertes.fr/tel-00932092 Submitted on 16 Jan 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE DE BOURGOGNE UFR Sciences et Techniques Institut de Math´ematiquesde Bourgogne THESE pour obtenir le grade de Docteur de l'Universit´ede Bourgogne Discipline : MATHEMATIQUES par Vincent Nolot Convexit´eset probl`emesde transport optimal sur l'espace de Wiener. Soutenue publiquement le 27 Juin 2013 devant le Jury compos´ede Bernard BONNARD Universit´ede Bourgogne (examinateur) Guillaume CARLIER Universit´eParis Dauphine (examinateur) Luigi DE PASCALE Universit´ede Pise (rapporteur) Shizan FANG Universit´ede Bourgogne (directeur de th`ese) Ivan GENTIL Universit´ede Lyon (examinateur) Nicolas PRIVAULT Universit´ede Singapour (rapporteur) 2 R´esum´een Fran¸cais L'objet de cette th`eseest d'´etudierla th´eoriedu transport optimal sur un espace de Wiener abstrait. Les r´esultatsqui se trouvent dans quatre principales parties, portent • Sur la convexit´ede l'entropie relative. On prolongera des r´esultatsconnus en dimension finie, sur l'espace de Wiener muni d'une norme uniforme, `a savoir que l'entropie relative est (au moins faiblement) 1−convexe le long des g´eod´esiquesinduites par un transport optimal sur l'espace de Wiener. • Sur les mesures `adensit´e logarithmiquement concaves. Le premier des r´esultatsimportants consiste `amontrer qu'une in´egalit´ede type Harnack est vraie pour le semi-groupe induit par une telle mesure sur l'espace de Wiener. Le second des r´esultatsobtenus nous fournit une in´egalit´een di- mension finie (mais ind´ependante de la dimension), contr^olant la diff´erence de deux applications de transport optimal. • Sur le probl`emede Monge. On s'int´eresseraau probl`emede Monge sur l'espace de Wiener, muni de plusieurs normes : des normes `avaleurs finies, ou encore la pseudo-norme de Cameron-Martin. • Sur l'´equationde Monge-Amp`ere.Gr^aceaux in´egalit´esobtenues pr´ec´edemment, nous serons en mesure de construire des solutions fortes de l'´equationde Monge-Amp`ere(induite par le co^utquadratique) sur l'espace de Wiener, sous de faibles hypoth`esessur les densit´esdes mesures consid´er´ees. Mots cl´es: transport optimal, probl`emede Monge, convexit´e,espace de Wiener, ´equation de Monge-Amp`ere,dimension infinie, mesure logarithmiquement concave. 3 4 Abstract in english The aim of this PhD is to study the optimal transportation theory in some abstract Wiener space. You can find the results in four main parts and they are about • The convexity of the relative entropy. We will extend the well known results in finite dimension to the Wiener space, endowed with the uniform norm. To be precise the relative entropy is (at least weakly) geodesically 1−convex in the sense of the optimal transportation in the Wiener space. • The measures with logarithmic concave density. The first important result consists in showing that the Harnack inequality holds for the semi-group induced by such a measure in the Wiener space. The second one provides us a finite dimensional and dimension-free inequality which gives estimate on the difference between two optimal maps. • The Monge Problem. We will be interested in the Monge Problem on the Wiener endowed with different norms: either some finite valued norms or the pseudo-norm of Cameron-Martin. • The Monge-Amp`ereequation. Thanks to the inequalities obtained above, we will be able to build strong solutions of the Monge-Amp`ere(those which are induced by the quadratic cost) equation on the Wiener space, provided the considered measures satisfy weak conditions. Key words: optimal transport, Monge problem, convexity, Wiener space, Monge- Amp`ereequation, infinite dimension, logarithmic concave measure. 5 6 Remerciements Mes remerciements pour l'accomplissement de ce travail s'adressent principalement `aShizan Fang, qui m'a supervis´e,conseill´e,orient´ependant ces trois ann´ees.Tout cela a toujours ´et´eaccompagn´ed'enthousiasme et d'encouragements, en particulier dans les moments difficiles. Je lui adresse toute ma reconnaissance. Ce travail n'aurait jamais vu le jour sans le soutien de Patrick Gabriel, qui a co-encadr´emon m´emoirede recherche en master. Patrick fait partie des personnes qui m'ont scientifiquement et humainement apport´ele plus, au sein du laboratoire. Je le remercie d'avoir partag´e sa grande ouverture d'esprit sur les math´ematiques,l'enseignement et bien au-del`a. J'ai le plaisir de remercier Nicolas Privault et Luigi De Pascale qui m'ont fait l'honneur de rapporter ma th`ese,et tout autant les autres membres de mon jury, Bernard Bonnard, Guillaume Carlier et Ivan Gentil. Leur expertise dans des do- maines vari´esest largement reconnue. Je tiens ´egalement `aremercier Robert McCann qui m'accueille `al'Universit´ede Toronto, en ce moment m^eme o`uj'´ecrisces lignes. Parce que faire une th`ese,c'est aussi parfois rencontrer au-del`ades math´ematiciens, des personnalit´esint´eressantes, ouvertes, qui n'h´esitent pas `aaider les jeunes chercheurs, et sans qui la motivation redescendrait trop vite; je tiens `aremercier Nicolas Juillet, pour m'avoir accueilli `aStrasbourg avec beaucoup de sympathie d`esle d´ebutde ma th`ese,ainsi que pour tous les autres bons moments que l'on a v´ecuaux conf´erenceso`u l'on se retrouvait. Thierry Champion qui m'a grandement encourag´edans mes travaux durant un colloque `aOrsay, puis dans nos rencontres Dijonaises. Pierre-Andr´eZitt dont l'humour n'est plus `ad´emontrer, qui ´etaitpr´esent pour les deux premi`eresann´ees de ma th`ese,a toujours ´et´ecurieux et `al'´ecoute. Merci `aBernard Bonnard pour les relations d'amiti´eque l'on a li´eestout au long de ces trois ann´ees.Je voudrais saluer mon demi-fr`erede th`ese,Camille Tardif qui est une personne aux grandes qualit´eshu- maines, et je ne regrette que le fait qu'il aie pass´eplus de temps `aStrasbourg plut^ot qu’`aDijon. Merci finalement aux membres de mon ´equipe, l'´equipe SPAN, pour les initiatives PodEx et tout le reste. Les conditions de travail que le staff de l'IMB a mises `adisposition ´etaient par- ticuli`erement ad´equates. Un grand merci aux agents d'entretien, notamment Aziz pour son sourire quotidien. Un grand merci aux secr´etairespour leur d´evouement, et plus sp´ecifiquement `aCaroline, qui s'est occup´eeavec attention de toutes mes mis- sions, et avec qui j'ai toujours eu beaucoup de plaisir `a´echanger des histoires plus ou moins amusantes. A elles s'ajoutent notre biblioth´ecairePierre et notre informaticien Francis, qui sont au coeur du bon fonctionnement du laboratoire. Trois ann´eesde vie commune avec les diff´erents doctorants et post-doctorants du laboratoire, avec qui on pouvait partager nos sentiments sur le travail de recherche. Ces impressions que l'on d´ecouvreau cours d'une th`eseet que les doctorants sont certainement les mieux `am^emede consid´erer. Merci `avous pour l'environnement agr´eableque vous avez cr´e´e,et j'esp`ereque notre association tant aim´eecontinuera son ascension. J'ai une pens´eeparticuli`ere`atous mes co-bureaux, et je ne citerai qu'eux (pour ne pas en oublier dautres) : Gautier, Gabriel, Pauline, Eglantine, Martin, Yi Shi et ce bon vieil Alvaro. Autant de personnes qui ont contribu´e`ace que le bureau 213 devienne l'un des plus embl´ematiquesdu laboratoire. 7 On ne devient pas docteur du jour au lendemain, mais apr`esune succession d'´ev`ene- ments, une longue poursuite des ´etudesqui demandent de la pers´ev´erance,et c'est pourquoi je n'oublie pas mes amis qui m'ont permis de m'´evader du monde des math´ematiqueset en particulier au cours de ces trois derni`eresann´ees. Une pens´ee particuli`ere`aGa¨etanavec qui j'ai fait toute ma scolarit´e`al'Universit´ede Bourgogne. Merci pour l'estime que tu as eue pour moi, cela m'a sans aucun doute encourag´edans mon parcours. A tous les autres, des pays de Langres, dijonais ou d'ailleurs pour les soir´eeset vacances emplies de joie et de bonne humeur. Au m^emeniveau je remercie chaque membre du club Langres Natation 52, avec qui j'ai nou´edes liens tr`esforts. Partenaires d'entra^ınements, de stages, de comp´etitions,merci ! Sous la tutelle de R´emy, quel bonheur de se retrouver dans l'eau avec vous pour souffrir physiquement, d´ecompresseret se vider la t^ete.Je ne remercierai jamais assez mon ami Jean Cote, qui m'a enlev´ece fardeau de responsabilit´esau club, afin d'accomplir au mieux mon travail de recherche et d'enseignement. Merci Jean pour tout ce que tu m'as appris sur tant de domaines diff´erents, en si peu de temps, et j'esp`ereque cela n'est pas fini. Je remercie ma famille, et notamment mes parents qui m'ont toujours pouss´eet m'ont `achaque fois donn´eles moyens de r´eussirmes ´etudes.
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