Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: 
\begin{equation}\left\{
\begin{aligned}
&\partial\_t u(t, x) = \frac{1}{2}\partial\_x^2 u(t, x) - U(u(t)) + \dot W(dt, dx), \\
&\partial\_x u(t, 0) = \partial\_x u(t, 1) = 0, \\
&u(0, x) = v(x). 
\end{aligned}\right.
\end{equation}
Here $U$ is a map from $L^2[0, 1]$ to $L^2[0, 1]$. We assume that $U$ is bounded and Lipschitz continuous, thus the solution exists and is continuous (let the initial condition be continuous in $x$). My question involves two aspects. 


(i) If $U = \frac{1}{2}DV$ for some Frechet differentiable real function $V$ on $L^2[0, 1]$ and $D$ is the Frechet derivative operator, then it is easy to expect that the infinite measure on $C[0, 1]$: 
$$\nu(\mathrm du) = \exp(-V(u))\mu(\mathrm du)$$
is invariant for $u(t)$, where $\mu$ is the measure of a process $\{w\_x\}\_{x \in [0, 1]}$ such that $w\_x - w\_0$ is a standard BM on $[0, 1]$ and $w\_0$ subjects to the Lebesgue measure on $\mathbb R$. However I can not find reference for this result (since it deals with an infinite invariant measure). Could anyone give me a reference (or a proof) for this? 


(ii) If $U = \frac{1}{2}DV + B$, where $V$ is as same as the above, and $B$ satisfies $E\_\nu \langle Df, B \rangle\_{L^2[0, 1]} = 0$ for all $f \in C\_b^1(H)$, would it change the invariant measure?. 


Remark: I know a reference [T. Funaki, Nagoya Math. J., 1983] which treats (i) for local nonlinear term, i.e., $V(u)$ is replaced with $\{V(u(x))\}\_{x \in [0, 1]}$. It is proved through a broken-line approximation. If no other results maybe I have to go in this way. 



The (infinite) invariant measure of an SPDE
I check it with the standard Garlerkin method and confirmed that it is right, in both cases (i) and (ii). 


Discribe the proof briefly (under (ii)): 


Take a CONS of $H$ as $h\_1 = 1$ and $h\_k(x) = \cos[(k-1)\pi x]$. 


First notice that if $V(u) = V^\dagger(\langle u, h\_1 \rangle, \ldots, \langle u, h\_N \rangle)$ and $\langle B(u), h\_k \rangle = B\_k^\dagger(\langle u, h\_1 \rangle, \ldots, \langle u, h\_N \rangle)$ for some $V^\dagger$, $B\_k^\dagger \in C\_b^2(\mathbb R^N)$ for $k \leq N$, and $B\_k^\dagger = 0$ for $k > N$, then it becomes an $N$-dimensional diffusion and a infinite-dimensional OU process, and the result is obvious. 


Then take $V\_N^\dagger$ and $B\_{N, k}^\dagger$ be the marginal expectation, i.e., for $x \in \mathbb R^N$ and $k \leq N$, 
$$V\_N^\dagger(x) = \int\_{\mathbb R^\infty} V\left(\sum\_{1}^N x\_kh\_k + \sum\_{N+1}^\infty y\_kh\_k\right)\prod\_{N+1}^\infty \Phi\_k(y\_k)dy\_k, $$
$$B\_{N, k}^\dagger(x) = e^{2V\_N^\dagger(x)} \int\_{\mathbb R^\infty} U\_k\left(\sum\_{1}^N x\_kh\_k + \sum\_{N+1}^\infty y\_kh\_k\right)\prod\_{N+1}^\infty \Phi\_k(y\_k)dy\_k, $$
where $U\_k = e^{-2V}\langle B, h\_k \rangle$ and $\Phi\_k(x) = \sqrt{\lambda\_k\pi^{-1}}e^{-\lambda\_kx^2}$. This finite-dimensional approximation protect the condition (ii), so we have the invariant measure, due to the previous step. Finally take the limit. 


