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I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However I am quite unfamiliar with the subject.
My question what is a good reference treating the following question:

If $W(t)$ is a cylindrical (or Q-)Wiener process with values in a Hilbert space H and $\mu_n$ as well as $\Sigma_n$ are a sequence of adapted to the filtration generated by $W(t)$ then under what circumstances does the solution to the SDEs $$ dX_n(t) = \mu_n(t,X_n(t))dt +\sigma_n(t,X_n(t))dW(t) $$ converge to the solution of the SDE $$ dX(t) = \lim_{n\mapsto \infty}\mu_n(t,X_n(t))dt +\lim_{n\mapsto \infty}\sigma_n(t,X_n(t))dW(t)? $$

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Check out Chapter 5 for projection-based methods for SDEs on Hilbert spaces:

Second Order PDE's in Finite and Infinite Dimension: A Probabilistic Approach Springer-Verlag Authors: Cerrai, Sandra ISBN: 9783540421368

Check out Chapter 10 of the following textbook for a basic intro to convergence of discretizations (including some projection-based methods):

An Introduction to Computational Stochastic PDEs Part of Cambridge Texts in Applied Mathematics AUTHORS:Gabriel J. Lord, Catherine E. Powell, Tony Shardlow ISBN: 9780521728522

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    $\begingroup$ I'm not looking for the convergence of a discritization though, just conditions under which I have convergence of continuous time SDEs $\endgroup$ – AIM_BLB Aug 20 '16 at 14:34
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    $\begingroup$ I added a reference to projection methods, which are the standard way to obtain convergence results for SDEs on Hilbert spaces. $\endgroup$ – Nawaf Bou-Rabee Aug 20 '16 at 15:13

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