The (infinite) invariant measure of an SPDE

Question

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.

Answer 1

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.