Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where $\partial_tB$ denotes the generalized time-derivative of a cylindrical Brownian motion $B$.
I've read, that the solution $u$ of $(1)$ can be considered as
- either a real-valued function of $(x,t)$ (let's call that the first view) or a function of $t$ with values in an infinite dimensional space of functions of $x$ (let's call that the second view), if $G=0$
- either a real-valued stochastic process indexed by $(x,t)$ which is the solution of a multiparameter SDE (first view) or a stochastic process indexed by $t$ with values in an infinite dimensional space of functions of $x$ which is the solution of an infinite dimensional SDE (second view), if $G\ne 0$
However, are both views really equivalent? If $G=0$, the first view implies that $u(\;\cdot\;,x)$ should be of class $C^1$, for any $x$. But in the second view, $u$ (which is a function of $t$ only in that case) doesn't need to be differentiable at all (in the classical sense).
Whether or not $G=0$, the choice of the infinite dimensional space of functions in the second view, seems to influence the solvability and uniqueness as well as the notion of a soluation of $(1)$.
So, is there anything I'm missing? What's the reason why we don't use the first view? And what's the "right choice" for the infinite dimensional space of functions?
[Please note, that I'm particular interested in an answer for the case $G\ne 0$]