, In this subsection we are again given a progressive path-dependent canonical class (P s,? ) (s,?)?R + ×? satisfying Hypothesis 6.3.4 and the corresponding path-dependent system of projectors (P s ) s?R +. Chapter 6. Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales, this subsection, all results come from Section 5.4 in Chapter 5

?. , A)

·. and M. ,

, We now consider the following abstract path-dependent non linear equation

·. A?-+-f-(·, ?. , ?. , and ?. ,

, It belongs to the space of symmetric matrices of size d and its coordinates will be denoted ( 2 i,j ? s (?)) i,j?d

, We say that ? is horizontally differentiable at (s, ?) ? ? T , s < T

, = lim s?T D? s (? s ) exists for every ? ? ? T. In this case, D? : (s, ?) ?? D? s (?) will be called the horizontal derivative of ?. If it is Borel, it defines a real valued functional. ? will be said continuous at fixed times if for all t, We say that ? is horizontally differentiable if it is horizontally differentiable in (s, ?) for all (s, ?) ? ? T such that s < T and the limit D? T (?)

, ? will be said boundedness preserving if for any compact set K of R d there exists a constant C K > 0 such that for all t ? [0, T ] and ? ? ? t taking values in K, we have |? t (?)| ? C K. ? will be

, T ) the space of real valued functionals ? constant after time T

, For every ? ? ?, and t ? 0, we denote by ? t ? the element of ? t defined by ? t ? (r) = ?(r) if r ? [0, t

, ,?)?R + ×? solves the well-posed martingale problem associated to (D(A), A), see Definition 6.3.17. 2. (D(A), A) is a weak generator of (P s ) s?R + , which is the unique path-dependent system of projectors for

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R. Zhu, Au delà de me laisser travailler à mon rythme et d'où je voulais, tu m'as petit à petit laissé faire mes propres choix de sujets, tout en restant d'excellent conseil. Je pense que dans mon cas précis, on aurait pas pu imaginer meilleur encadrement. Merci à mes rapporteur.e.s pour leur relecture attentive et à mon Jury de soutenance de m'avoir accordé sa confiance et d'avoir posé des questions qui me permettront de pousser mes travaux plus loin. Je remercie aussi David et Olivier, avec qui ça a été (et ça continue d'être puisqu'on ne change pas une équipe qui gagne) un plaisir d'enseigner, et de découvrir que j'aimais enseigner. Une pensée à Christophe dont le sourire permanent est une valeur sûre et qui m'a aidé à me mettre à Linux, ainsi qu'à Corinne pour ses excellents conseils de lecture, et qui a sû me supporter malgré mon incapacité totale à conserver mes reçus de paiement en déplacement, ou à noter mes vacances sur l'intranet. Je remercie bien sûr toute l'UMA qui m'a très bien accueilli, bien qu'il faille reconnaître que j'ai été très peu présent dans la vie du labo. Le fait d'avoir des amis de longue date à l'X avec qui déjeuner y était pour quelque chose. Merci à ma famille. À ma mère qui en a vu de toutes les couleurs pendant ces trois ans et qui a eu le toupet de réussir ses exams avant moi. Ma marraine, mon parrain et mes soeurs de coeur qui ont été la dans les moments durs, mais que j'ai aussi eu la chance d'accompagner quand leur famille s'agrandissait. Mon beau-père et ses fils, Remerciements Avant tout je voudrais remercier Francesco. Il est certain que tout.e élève remercie ici son directeur ou sa directrice de thèse avec plus ou moins de sincérité, vol.15, p.1250022, 2012.

, De NDDL à Bure, en passant par Ispahan, Pontpoint ou Bisca, on s'est pas laissé abattre. À tous ces dragons cachés dans le placard et toutes ces découvertes faites ensemble. À ma doctorante russe préférée qui a accompli l'exploit de faire d'Anthony, Palaiseau ou Orsay des villes dont je garderai de bons souvenirs; à mes ami.e.s de longue date; de teufs interminables qui nous laissaient le champ libre mais sur les quelles on ne fera pas de zoom, Merci à la colloc de l'amour, pour ces années qu'on ne peut résumer en quelques lignes, vous même vous savez

, aux zadistes, aux féministes, aux écologistes, aux antifascites, aux antiracistes, aux victimes des violences policières et leurs familles, aux prisonnier.e.s politiques ou non, aux situationistes, aux squateur.euse.s, aux bloqueur.euse.s, aux hiboux, aux communistes libertaires, aux mutuelistes libertaires, aux collectivistes libertaires, aux anarchosyndicalistes, aux collectifs de sans-papiers, à ceux de mal logé.e.s, aux antipubs, aux libristes, aux vegans, au cortège de tête, aux casseur.euse.s, aux grévistes, à toutes celles et ceux qui s'organisent, Je termine par une pensée à mes camarades. Aux anarchistes, aux libertaires, aux syndicalistes révolutionnaires, aux autonaumes