, 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

C. D. Aliprantis and K. C. Border, Infinite-dimensional analysis

D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc, vol.73, pp.890-896, 1967.

D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators, vol.348, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00929960

V. Bally, E. Pardoux, and L. Stoica, Backward stochastic differential equations associated to a symmetric Markov process, Potential Anal, vol.22, issue.1, pp.17-60, 2005.
URL : https://hal.archives-ouvertes.fr/inria-00072163

E. Bandini, Existence and uniqueness for backward stochastic differential equations driven by a random measure, Electronic Communications in Probability, vol.20, issue.71, pp.1-13, 2015.

E. Bandini and F. Russo, Special weak Dirichlet processes and BSDEs driven by a random measure, Bernoulli, vol.24, issue.4A, pp.2569-2609, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01241076

G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential equations and integralpartial differential equations, Stochastics: An International Journal of Probability and Stochastic Processes, vol.60, pp.57-83, 1997.

G. Barles, E. Lesigne, . Sde, and P. Bsde, Backward stochastic differential equations, vol.364, pp.47-80, 1995.

A. Barrasso and F. Russo, Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated P seudo Partial Differential Equations, p.1431559, 2017.

A. Barrasso and F. Russo, Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. part II: Decoupled mild solutions and examples, p.1505974, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01431559

A. Barrasso and F. Russo, Martingale driven BSDEs, PDEs and other related deterministic problems, p.1566883, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01566883

A. Barrasso and F. Russo, A note on time-dependent additive functionals, Communications on Stochastic Analysis, vol.11, p.2017
URL : https://hal.archives-ouvertes.fr/hal-01574964

A. Barrasso and F. Russo, BSDEs and decoupled mild solutions of path-dependent PDEs and IPDEs, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01774823

A. Barrasso and F. Russo, Path-dependent Martingale Problems and Additive Functionals, p.1775200, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01775200

A. Ben-israel and T. N. Greville, Generalized inverses, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol.15

P. Billingsley, Probability and measure, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, 1986.

J. Bion, Dynamic risk reasures and path-dependent second order PDEs, Stochastics of Environmental and Financial Economics, vol.138, pp.147-178, 2016.

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl, vol.44, pp.384-404, 1973.

R. M. Blumenthal, R. K. Getoor, and H. P. Mckean, Markov processes with identical hitting distributions, Bull. Amer. Math. Soc, vol.68, pp.372-373, 1962.

R. Buckdahn, Backward stochastic differential equations driven by a martingale. Unpublished, 1993.

G. Cannizzaro and K. Chouk, Multidimensional sdes with singular drift and universal construction of the polymer measure with white noise potential, 2015.

R. Carbone, B. Ferrario, and M. Santacroce, Backward stochastic differential equations driven by càdlàg martingales, Teor. Veroyatn. Primen, vol.52, issue.2, pp.375-385, 2007.

C. Ceci, A. Cretarola, and F. Russo, BSDEs under partial information and financial applications, Stochastic Process. Appl, vol.124, issue.8, pp.2628-2653, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00822988

F. Confortola, M. Fuhrman, and J. Jacod, Backward stochastic differential equation driven by a marked point process: an elementary approach with an application to optimal control, Ann. Appl. Probab, vol.26, issue.3, pp.1743-1773, 2016.

R. Cont and D. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal, vol.259, issue.4, pp.1043-1072, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00471318

R. Cont and D. Fournié, A functional extension of the Itô formula, C. R. Math. Acad. Sci, vol.348, issue.1-2, pp.57-61, 2010.

R. Cont and D. Fournié, Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab, vol.41, issue.1, pp.109-133, 2013.

A. Cosso and F. Russo, Strong-viscosity solutions: semilinear parabolic PDEs and path-dependent PDEs, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01145301

A. Cosso and F. Russo, Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations, Infin. Dimens. Anal. Quantum Probab. Relat. Top, vol.19, issue.4, p.44, 2016.

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions, of Encyclopedia of Mathematics and its Applications, vol.152, 2014.

F. Delarue and R. Diel, Rough paths and 1d SDE with a time dependent distributional drift: application to polymers, vol.165, pp.1-63, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00947201

C. Dellacherie and P. Meyer, Probabilités et potentiel, volume A. Hermann, Paris, 1975. Chapitres I à IV

C. Dellacherie and P. Meyer, Probabilités et potentiel. Chapitres V à VIII, volume 1385 of Actualités Scientifiques et Industrielles

P. Hermann, Théorie des martingales, 1980.

C. Dellacherie and P. Meyer, Probabilités et potentiel. Chapitres XII-XVI, 1987.

C. , D. Girolami, and F. Russo, Infinite dimensional stochastic calculus via regularization and applications, 2010.

E. Di-nezza, G. Palatucci, and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math, vol.136, issue.5, pp.521-573, 2012.

B. Dupire, Functional Itô calculus. Portfolio Research Paper, 2009.

E. B. Dynkin, Osnovaniya teorii markovskikh protsessov. Teorija Verojatnoste? ? i Matemati?eskaja Statistika, Gosudarstv. Izdat. Fiz.-Mat. Lit, 1959.

E. B. Dynkin, Additive functionals of markov processes and stochastic systems, Annales de l'institut Fourier, vol.25, pp.177-200, 1975.

E. B. Dynkin, Markov processes and related problems of analysis, London Mathematical Society Lecture Note Series, vol.54, 1982.

A. Einstein, Investigations on the theory of the Brownian movement, 1956.

I. Ekren, C. Keller, N. Touzi, and J. Zhang, On viscosity solutions of path dependent PDEs, Ann. Probab, vol.42, issue.1, pp.204-236, 2014.

N. E. Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical finance, vol.7, issue.1, pp.1-71, 1997.

S. N. Ethier and T. G. Kurtz, Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, 1986.

G. Fabbri, F. Gozzi, and A. Swi-&apos;-ech, Stochastic optimal control in infinite dimension, of Probability Theory and Stochastic Modelling, vol.82, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01447562

F. Flandoli, E. Issoglio, and F. Russo, Multidimensional stochastic differential equations with distributional drift, Trans. Amer. Math. Soc, vol.369, issue.3, pp.1665-1688, 2017.
URL : https://hal.archives-ouvertes.fr/hal-00935399

F. Flandoli, F. Russo, and J. Wolf, Some SDEs with distributional drift. I. General calculus, Osaka J. Math, vol.40, issue.2, pp.493-542, 2003.

F. Flandoli, F. Russo, and J. Wolf, Some SDEs with distributional drift. II. Lyons-Zheng structure, Itô's formula and semimartingale characterization, Random Oper. Stochastic Equations, vol.12, issue.2, pp.145-184, 2004.

F. Flandoli and G. Zanco, An infinite-dimensional approach to path-dependent Kolmogorov equations, Ann. Probab, vol.44, issue.4, pp.2643-2693, 2016.

M. Fuhrman, F. Masiero, and G. Tessitore, Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim, vol.48, issue.7, pp.4624-4651, 2010.

M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, Ann. Probab, vol.30, issue.3, pp.1397-1465, 2002.

M. Fuhrman and G. Tessitore, Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations, Appl. Math. Optim, vol.51, issue.3, pp.279-332, 2005.

M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, 1994.
DOI : 10.1515/9783110889741

W. Hoh, Pseudo differential operators generating markov processes. Habilitations-schrift, Universität Bielefeld, 1998.

E. P. Hsu, Stochastic analysis on manifolds, vol.38, 2002.

E. Issoglio and S. Jing, Forward-Backward SDEs with distributional coefficients, 2016.

N. Jacob, Pseudo differential operators and Markov processes, vol.I, 2001.
DOI : 10.1142/p395

N. Jacob, Pseudo differential operators & Markov processes, Generators and their potential theory, vol.II, 2002.
DOI : 10.1142/p395

N. Jacob, Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, vol.3, 2005.
DOI : 10.1142/p395

J. Jacod, Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics, vol.714, 1979.
DOI : 10.1007/bfb0064907

J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften, vol.288

. Springer-verlag, , 2003.

J. Jost, Riemannian Geometry and Geometric Analysis, 2011.
DOI : 10.1007/978-3-319-61860-9

URL : http://cds.cern.ch/record/1666885/files/9783540773405_TOC.pdf

T. Klimsiak, Semi-Dirichlet forms, Feynman-Kac functionals and the Cauchy problem for semilinear parabolic equations, J. Funct. Anal, vol.268, issue.5, pp.1205-1240, 2015.

I. Laachir and F. Russo, BSDEs, càdlàg martingale problems, and orthogonalization under basis risk, SIAM J. Financial Math, vol.7, issue.1, pp.308-356, 2016.
DOI : 10.1137/140996239

D. Leão, A. Ohashi, and A. B. Simas, Weak functional Itô calculus and applications, 2014.

G. Liang, T. Lyons, and Z. Qian, Backward stochastic dynamics on a filtered probability space, Ann. Probab, vol.39, issue.4, pp.1422-1448, 2011.
DOI : 10.1214/10-aop588

URL : https://doi.org/10.1214/10-aop588

G. Liang, T. Lyons, and Z. Qian, Backward stochastic dynamics on a filtered probability space, Ann. Probab, vol.39, issue.4, pp.1422-1448, 2011.

P. Meyer, Fonctionelles multiplicatives et additives de Markov, Ann. Inst. Fourier (Grenoble), vol.12, pp.125-230, 1962.
DOI : 10.5802/aif.121

URL : http://www.numdam.org/article/AIF_1962__12__125_0.pdf

P. A. Meyer, Séminaire de Probabilités, X, Seconde partie: Théorie des intégrales stochastiques, vol.511, 1976.

É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Stochastic analysis and related topics, vol.42, pp.79-127, 1996.

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett, vol.14, issue.1, pp.55-61, 1990.

É. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic partial differential equations and their applications, vol.176, pp.200-217, 1991.
DOI : 10.1007/bfb0007334

É. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic partial differential equations and their applications, vol.176, pp.200-217, 1991.
DOI : 10.1007/bfb0007334

E. Pardoux and A. , Stochastic differential equations, backward SDEs, partial differential equations, Stochastic Modelling and Applied Probability, vol.69, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01108223

K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, issue.3, 1967.

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep, vol.37, issue.1-2, pp.61-74, 1991.

S. Peng and F. Wang, BSDE, path-dependent PDE and nonlinear Feynman-Kac formula, Science China Mathematics, vol.59, issue.1, pp.19-36, 2016.
DOI : 10.1007/s11425-015-5086-1

URL : http://arxiv.org/pdf/1108.4317

P. E. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability, vol.21, 2004.

J. P. Roth, Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues, Ann. Inst. Fourier (Grenoble), vol.26, issue.4, pp.1-97, 1976.

A. Rozkosz, Weak convergence of diffusions corresponding to divergence form operators, Stochastics Stochastics Rep, vol.57, issue.1-2, pp.129-157, 1996.

F. Russo and G. Trutnau, Some parabolic PDEs whose drift is an irregular random noise in space, Ann. Probab, vol.35, issue.6, pp.2213-2262, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00019856

F. Russo and L. Wurzer, Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time, Stochastics and Dynamics, 2015.

M. Schweizer, Approximating random variables by stochastic integrals, Ann. Probab, vol.22, issue.3, pp.1536-1575, 1994.

D. W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol.32, issue.3, pp.209-244, 1975.

D. W. Stroock and S. R. Varadhan, Multidimensional diffusion processes, Classics in Mathematics, 2006.

F. Viens and J. Zhang, A Martingale Approach for Fractional Brownian Motions and Related Path Dependent PDEs, 2017.

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