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Suppose: $$ dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t $$ is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $X_t$ in terms of a basis $\{e_i\}$ of $H$?

That is, if $\{e_i\}$ is a basis for $H$ then is the previous SDE equivalent to: $$ dX_t= \sum_i dX_te_i =\sum_i \left(a(t,X_t)e_idt + b(t,X_t)e_idW_t^H \right) ? $$

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In principle, the variational form of the SPDE problem allows you to do this. What you do is expand the solution in terms of basis vectors, and choose the test functions in the variational formulation to be basis vectors. This is how projection-based methods for SPDEs are constructed.

For more detail see, e.g., Definition 10.19 and Section 10.6 of: An Introduction to Computational Stochastic PDEs Part of Cambridge Texts in Applied Mathematics by Gabriel J. Lord, Catherine E. Powell, Tony Shardlow

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  • $\begingroup$ Perfect! Do you have a link to a reference that outlines how to do this? I am unfamiliar with this literature $\endgroup$
    – ABIM
    Commented Aug 27, 2016 at 12:56
  • $\begingroup$ I added a reference $\endgroup$
    – Nawaf Bou-Rabee
    Commented Aug 27, 2016 at 13:07
  • $\begingroup$ Excellent! I actually have that book but never browsed that far in, thank you very much! $\endgroup$
    – ABIM
    Commented Aug 27, 2016 at 13:12
  • $\begingroup$ I'll try to add a few more references to this variational approach next week. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Aug 27, 2016 at 13:40

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