R. Buckdahn, J. Ma, and J. Zhang, Pathwise Taylor expansions for random fields on multiple dimensional paths, Stochastic Processes and their Applications, vol.125, issue.7, 2013.
DOI : 10.1016/j.spa.2015.02.004

S. Cerrai, Second order PDE's in finite and infinite dimension, Lecture Notes in Mathematics, vol.1762, 2001.
DOI : 10.1007/b80743

S. Cerrai and G. Da-prato, Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction???diffusion equations, Journal of Evolution Equations, vol.118, issue.1, pp.83-98, 2012.
DOI : 10.1007/s00028-011-0124-0

M. C. Cerutti, L. Escauriaza, and E. B. Fabes, Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients, Annali di Matematica Pura ed Applicata, vol.96, issue.no. 4, pp.161-180, 1993.
DOI : 10.1007/BF01759020

A. and C. Michalik, Representation theorem for general stochastic delay equations, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys, vol.26, pp.635-642, 1978.

R. Cont and D. Fournié, Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, vol.259, issue.4, pp.1043-1072, 2010.
DOI : 10.1016/j.jfa.2010.04.017

URL : https://hal.archives-ouvertes.fr/hal-00471318

R. Cont and D. Fournié, A functional extension of the Ito formula, Comptes Rendus Mathematique, vol.348, issue.1-2, pp.57-61, 2010.
DOI : 10.1016/j.crma.2009.11.013

URL : https://hal.archives-ouvertes.fr/hal-00457533

R. Cont and D. Fournié, Functional It?? calculus and stochastic integral representation of martingales, The Annals of Probability, vol.41, issue.1, pp.109-133, 2013.
DOI : 10.1214/11-AOP721

A. Cosso and F. Russo, Kolmogorov equations for path-dependent SDEs

A. Cosso and F. Russo, A regularization approach to functional Itô calculus and strong-viscosity solutions to path-dependent PDEs, 2014.

M. G. Crandall, M. Kocan, P. Soravia, and A. Swi-'ech, On the equivalence of various weak notions of solutions of elliptic PDEs with measurable ingredients, Progress in elliptic and parabolic partial differential equations (Capri, pp.136-162, 1994.

G. , D. Prato, and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, Journal of Functional Analysis, vol.259, issue.1, pp.243-267, 2010.

G. Da-prato and J. Zabczyk, Stochastic equations in infinite dimensions, volume 44 of Encyclopedia of Mathematics and its Applications, 1992.

G. Da-prato and J. Zabczyk, Second order partial differential equations in Hilbert spaces, 2002.
DOI : 10.1017/CBO9780511543210

C. , D. Girolami, G. Fabbri, and F. Russo, The covariation for Banach space valued processes and applications, Metrika, vol.77, issue.1, pp.51-104, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00780430

D. Girolami and F. Russo, About Kolmogorov equations associated to window processes, preparation, 2014.

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

C. , D. Girolami, and F. Russo, Clark-Ocone type formula for non-semimartingales with finite quadratic variation, C. R. Math. Acad. Sci. Paris, vol.349, pp.3-4209, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00484993

D. Girolami and F. Russo, GENERALIZED COVARIATION AND EXTENDED FUKUSHIMA DECOMPOSITION FOR BANACH SPACE-VALUED PROCESSES: APPLICATIONS TO WINDOWS OF DIRICHLET PROCESSES, Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol.15, issue.02, p.1250007, 2012.
DOI : 10.1142/S0219025712500075

C. , D. Girolami, and F. Russo, Generalized covariation for Banach space valued processes, Itô formula and applications, Osaka Journal of Mathematics, vol.51, issue.3, p.2014

N. Dinculeanu, Vector integration and stochastic integration in Banach spaces, Pure and Applied Mathematics, 2000.

B. Dupire, Functional It?? Calculus, SSRN Electronic Journal, 2009.
DOI : 10.2139/ssrn.1435551

I. Ekren, . Ch, N. Keller, J. Touzi, and . Zhang, On viscosity solutions of path dependent PDEs, The Annals of Probability, vol.42, issue.1, pp.204-236, 2014.
DOI : 10.1214/12-AOP788

I. Ekren, N. Touzi, and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I, 2013.

I. Ekren, N. Touzi, and J. Zhang, Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II, The Annals of Probability, vol.44, issue.4, 2013.
DOI : 10.1214/15-AOP1027

I. Ekren, N. Touzi, and J. Zhang, Optimal stopping under nonlinear expectation. Stochastic Process, Appl, vol.124, issue.10, pp.3277-3311, 2014.
DOI : 10.1016/j.spa.2014.04.006

URL : http://arxiv.org/abs/1209.6601

F. Flandoli and F. Gozzi, Kolmogorov Equation Associated to a Stochastic Navier???Stokes Equation, Journal of Functional Analysis, vol.160, issue.1, pp.312-336, 1998.
DOI : 10.1006/jfan.1998.3321

F. Flandoli and G. Zanco, An infinite-dimensional approach to path-dependent Kolmogorov's equations, 2013.

M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, The Annals of Probability, vol.30, issue.3, pp.1397-1465, 2002.
DOI : 10.1214/aop/1029867132

F. Gozzi and F. Russo, Verification theorems for stochastic optimal control problems via a time dependent Fukushima-Dirichlet decomposition. Stochastic Process, Appl, vol.116, issue.11, pp.1530-1562, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00022840

F. Gozzi and F. Russo, Weak Dirichlet processes with a stochastic control perspective. Stochastic Process, Appl, vol.116, issue.11, pp.1563-1583, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00022839

M. Gradinaru and I. Nourdin, Approximation at First and Second Order of $m$-order Integrals of the Fractional Brownian Motion and of Certain Semimartingales, Electronic Journal of Probability, vol.8, issue.0, p.pp, 2003.
DOI : 10.1214/EJP.v8-166

URL : https://hal.archives-ouvertes.fr/hal-00091322

R. Jensen, Uniformly Elliptic PDEs with Bounded, Measurable Coefficients, Journal of Fourier Analysis and Applications, vol.2, issue.3, pp.237-259, 1996.
DOI : 10.1007/s00041-001-4031-6

R. Jensen, M. Kocan, and A. Swi-'ech, Good and viscosity solutions of fully nonlinear elliptic equations, Proc. Amer, pp.533-542, 2002.

G. Kallianpur, I. Mitoma, and R. L. Wolpert, Diffusion equations in duals of nuclear spaces, Stochastics An International Journal of Probability and Stochastic Processes, vol.29, issue.2, pp.285-329, 1990.
DOI : 10.1080/17442509008833618

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

F. Masiero, Stochastic Optimal Control Problems and Parabolic Equations in Banach Spaces, SIAM Journal on Control and Optimization, vol.47, issue.1, pp.251-300, 2008.
DOI : 10.1137/050632725

M. Métivier, Semimartingales: a course on stochastic processes, De Gruyter Studies in Mathematics, vol.2, 1982.
DOI : 10.1515/9783110845563

M. Métivier and J. Pellaumail, Stochastic integration, 1980.

S. E. Mohammed, Stochastic functional differential equations, Research Notes in Mathematics, vol.99, 1984.

S. Peng, Note on viscosity solution of path-dependent PDE and G-martingales, 2012.

Z. Ren, N. Touzi, and J. Zhang, An Overview of Viscosity Solutions of Path-Dependent PDEs, 2014.
DOI : 10.1007/978-3-319-11292-3_15

URL : https://hal.archives-ouvertes.fr/hal-01057300

M. Röckner and Z. Sobol, A new approach to Kolmogorov equations in infinite dimensions and applications to the stochastic 2D Navier???Stokes equation, Comptes Rendus Mathematique, vol.345, issue.5, pp.289-292, 2007.
DOI : 10.1016/j.crma.2007.07.009

F. Russo and P. Vallois, Intégrales progressive, rétrograde et symétrique de processus non adaptés, C. R. Acad. Sci. Paris Sér. I Math, issue.8, pp.312615-618, 1991.

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields, pp.403-421, 1993.
DOI : 10.1007/bf01195073

F. Russo and P. Vallois, The generalized covariation process and Itô formula. Stochastic Process, Appl, vol.59, issue.1, pp.81-104, 1995.

F. Russo and P. Vallois, Stochastic calculus with respect to continuous finite quadratic variation processes, Stochastics An International Journal of Probability and Stochastic Processes, vol.70, issue.1, pp.1-40, 2000.
DOI : 10.1080/17442500008834244

F. Russo and P. Vallois, Elements of Stochastic Calculus via Regularization, Séminaire de Probabilités XL, pp.147-185, 2007.
DOI : 10.1007/978-3-540-71189-6_7

S. Tang and F. Zhang, Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations, Discrete and Continuous Dynamical Systems, vol.35, issue.11, 2013.
DOI : 10.3934/dcds.2015.35.5521

J. M. Van-neerven, M. C. Veraar, and L. Weis, Stochastic integration in UMD Banach spaces, The Annals of Probability, vol.35, issue.4, pp.1438-1478, 2007.
DOI : 10.1214/009117906000001006

J. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math, vol.1180, pp.265-439
DOI : 10.1007/BFb0074920