R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, vol.65, 1975.

E. Alòs, D. Nualart, and F. Viens, Stochastic heat equation with white-noise drift, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.36, issue.2, pp.181-218, 2000.
DOI : 10.1016/S0246-0203(00)00122-9

A. Andersson, M. Hefter, A. Jentzen, and R. Kurniawan, Regularity Properties for Solutions of Infinite Dimensional Kolmogorov Equations in Hilbert Spaces, Potential Analysis, vol.398, issue.1, 2016.
DOI : 10.1007/BFb0092419

A. Andersson, R. Kruse, and S. Larsson, Duality in refined Sobolev???Malliavin spaces and weak approximation of SPDE, Stochastics and Partial Differential Equations Analysis and Computations, vol.43, issue.1, pp.113-149, 2016.
DOI : 10.1137/040605278

A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation, Mathematics of Computation, vol.85, issue.299, pp.1335-1358, 2016.
DOI : 10.1090/mcom/3016

V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields, pp.43-60, 1996.
URL : https://hal.archives-ouvertes.fr/inria-00074016

C. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise. Potential Anal, pp.1-40, 2014.

C. Bréhier, Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs, 2017.

C. Bréhier and M. Kopec, Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme, IMA Journal of Numerical Analysis, vol.43, 2013.
DOI : 10.1137/040605278

Z. Brze?niak, On stochastic convolution in banach spaces and applications, Stochastics and Stochastic Reports, vol.1180, issue.3-4, pp.245-295, 1997.
DOI : 10.1007/978-94-009-3873-1

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

E. Clément, A. Kohatsu-higa, and D. Lamberton, A duality approach for the weak approximation of stochastic differential equations, The Annals of Applied Probability, vol.16, issue.3, pp.1124-1154, 2006.
DOI : 10.1214/105051606000000060

D. Conus, A. Jentzen, and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. arxiv:1408, 1108.

S. Cox, A. Jentzen, R. Kurniawan, and P. Pu?nik, On the mild It\ô formula in Banach spaces, 2016.

G. and D. Prato, Kolmogorov equations for stochastic PDEs Advanced Courses in Mathematics, 2004.

G. and D. Prato, Kolmogorov Equations for Stochastic PDE's with Multiplicative Noise, Stochastic analysis and applications, pp.235-263, 2007.
DOI : 10.1007/978-3-540-70847-6_10

G. Da-prato, A. Jentzen, and M. Röckner, A mild Itô formula for SPDEs. arXiv preprint arXiv:1009, 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, volume 293 of Lecture Notes, 2002.

A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Mathematics of Computation, vol.80, issue.273, pp.89-117, 2011.
DOI : 10.1090/S0025-5718-2010-02395-6
URL : https://www.ams.org/mcom/2011-80-273/S0025-5718-2010-02395-6/S0025-5718-2010-02395-6.pdf

A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation, Mathematics of Computation, vol.78, issue.266, pp.845-863, 2009.
DOI : 10.1090/S0025-5718-08-02184-4
URL : https://hal.archives-ouvertes.fr/hal-00183249

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Mathematics of Computation, vol.58, issue.198, pp.603-630, 1992.
DOI : 10.1090/S0025-5718-1992-1122067-1
URL : http://www.ams.org/mcom/1992-58-198/S0025-5718-1992-1122067-1/S0025-5718-1992-1122067-1.pdf

F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by??stochastic perturbation, Inventiones mathematicae, vol.93, issue.135, pp.1-53, 2010.
DOI : 10.1070/SM1974v022n01ABEH001689
URL : https://hal.archives-ouvertes.fr/hal-00359723

M. Geissert, M. Kovács, and S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT Numerical Mathematics, vol.44, issue.2, pp.343-356, 2009.
DOI : 10.1090/S0025-5718-08-02184-4

I. C. Gohberg and M. G. , Kre? ? n. Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A, Feinstein. Translations of Mathematical Monographs, vol.18, 1969.

M. Hairer, A. M. Stuart, and J. Voss, Analysis of SPDEs arising in path sampling part II: The nonlinear case, The Annals of Applied Probability, vol.17, issue.5/6, pp.5-61657, 2007.
DOI : 10.1214/07-AAP441
URL : http://doi.org/10.1214/07-aap441

E. Hausenblas, Weak approximation for semilinear stochastic evolution equations In Stochastic analysis and related topics VIII, Progr. Probab, vol.53, pp.111-128, 2003.
DOI : 10.1007/978-3-0348-8020-6_5

M. Hefter, A. Jentzen, and R. Kurniawan, Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces. arXiv preprint, 2016.

L. Jacobe-de-naurois, A. Jentzen, and T. Welti, Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise, 2015.

A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, 2015.

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations springer-verlag, 1992.

M. Kovács, S. Larsson, and F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise, BIT Numerical Mathematics, vol.44, issue.1, pp.85-108, 2012.
DOI : 10.1007/978-1-4612-6027-1

M. Kovács, S. Larsson, and F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes, BIT Numerical Mathematics, vol.50, issue.2, pp.497-525, 2013.
DOI : 10.1007/978-1-4612-6027-1

N. V. Krylov, M. Röckner, and J. Zabczyk, Stochastic PDE's and Kolmogorov equations in infinite dimensions, volume 1715 of Lecture Notes in Mathematics Lectures given at the 2nd C.I.M.E. Session held in Cetraro, I.M.E. Foundation], 1998.

J. A. León and D. Nualart, Stochastic evolution equations with random generators, Ann. Probab, vol.26, issue.1, pp.149-186, 1998.

G. N. Milstein and M. V. Tretyakov, Stochastic numerics for mathematical physics. Scientific Computation, 2004.
DOI : 10.1007/978-3-662-10063-9

D. Nualart, The Malliavin calculus and related topics. Probability and its Applications, 2006.
DOI : 10.1007/978-1-4757-2437-0

D. Nualart and É. Pardoux, Stochastic calculus with anticipating integrands. Probab. Theory Related Fields, pp.535-581, 1988.
DOI : 10.1007/bf00353876

D. Nualart and F. Viens, Evolution equation of a stochastic semigroup with white-noise drift, The Annals of Probability, vol.28, issue.1, pp.36-73, 2000.
DOI : 10.1214/aop/1019160111
URL : https://doi.org/10.1214/aop/1019160111

É. Pardoux, Two-sided stochastic calculus for spdes, Stochastic partial differential equations and applications, pp.200-207, 1985.
DOI : 10.1137/1120030

É. Pardoux and P. Protter, A two-sided stochastic integral and its calculus. Probab. Theory Related Fields, pp.15-49, 1987.
DOI : 10.1007/bf00390274

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol.44, 1983.
DOI : 10.1007/978-1-4612-5561-1

J. Printems, On the discretization in time of parabolic stochastic partial differential equations Monte Carlo Methods Appl Monte Carlo and probabilistic methods for partial differential equations, pp.359-368, 2000.

M. Röckner and Z. Sobol, Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations, The Annals of Probability, vol.34, issue.2, pp.663-727, 2006.
DOI : 10.1214/009117905000000666

D. W. Strook and S. R. Varadhan, Multidimensional diffusion processes, Lecture Notes in Mathematics, 2006.
DOI : 10.1007/3-540-28999-2

D. Talay, Discr??tisation d'une ??quation diff??rentielle stochastique et calcul approch?? d'esp??rances de fonctionnelles de la solution, ESAIM: Mathematical Modelling and Numerical Analysis, vol.20, issue.1, pp.141-179, 1986.
DOI : 10.1051/m2an/1986200101411
URL : https://www.esaim-m2an.org/articles/m2an/pdf/1986/01/m2an1986200101411.pdf

A. Tambue and J. M. Ngnotchouye, Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise, Applied Numerical Mathematics, vol.108, pp.57-86, 2016.
DOI : 10.1016/j.apnum.2016.04.013

H. , Interpolation theory, function spaces, differential operators, 1995.

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. M. Van-neerven, M. C. Veraar, and L. Weis, Stochastic evolution equations in UMD Banach spaces, Journal of Functional Analysis, vol.255, issue.4, pp.940-993, 2008.
DOI : 10.1016/j.jfa.2008.03.015

X. Wang, Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus, Discrete and Continuous Dynamical Systems, vol.36, issue.1, pp.481-497, 2016.
DOI : 10.3934/dcds.2016.36.481

X. Wang and S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, Journal of Mathematical Analysis and Applications, vol.398, issue.1, pp.151-169, 2013.
DOI : 10.1016/j.jmaa.2012.08.038