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
$$
dX(t) = \lim\_{n\mapsto \infty}\mu\_n(t,X\_n(t))dt +\lim\_{n\mapsto \infty}\sigma\_n(t,X\_n(t))dW(t)?
$$



Reference for convergence of Hilbert-space valued SDEs
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


