## Stochastic PDE, infinite dimensional Hilbert Space [HARD]

I am studying the following stochastic PDE (from Damiano Brigo's book):
$dl(t) = [Gl(t)+\beta(l(t))]dt + \sum_{k} {\gamma}^k (l(t))dW_t^k$
where ${l(t)}_{t\geq 0}$ is a stochastic process taking values in a Hilbert space H. G is the generator (of the strongly continuous semi-group of shifts) and $W^k$ is an infinite sequence of Brownian motions. One can take H to be the space of absolutely continuous functions from R_+ to R.

Here is my question: how do we address the question that each path maps the time-interval into an infinite dimensional Hilbert space ?! That is, the stochastic PDE generates an infinite dimensional path measure?

I think this is quite a challenging question.