Suppose $E$ is a locally compact Polish space, and $(X_t)_{t\ge 0}$ is a Markov process on $E$ with a Feller transition semi-group $P_t:C_b(E)\to C_b(E)$ with a stationary or even ergodic probability measure $\nu$ on $E$. Then we have $\mathbb{E}_\nu[f(X_t)]= \mathbb{E}_\nu[f(X_0)] = \int_E f(x)\nu(dx)$ for all $t\ge 0$.
However, stationary measures $\nu$ are in general hard to identify, and I am interested in finding a measure $\nu$ so that $\mathbb{E}_\nu[f(X_t)] = \mathbb{E}_\nu[f(X_0)]$ for a fixed $f$. My question is, is there a general method to find such $\nu$ for a fixed $f$? Or does such $\nu$ even exist?
Here is an example I have in mind: Let $D\subset \mathbb{R}^n$ be a compact domain with smooth boundary for $n \ge 1$, and let $X_t$ be the reflected Brownian motion on $D$ with normal reflection. This means, if we let $n(x)$ be the inward normal vector on $\partial D$, and $L_t$ be the boundary local time of $X_t$ on $\partial D$, then $X_t$ has the following Skorohod representation (cf.Fukashima, Oshima, Takeda 2010, p255): $$ X_t = X_0 +B_t + \frac{1}{2} \int_0^t n(X_s)dL_S,\quad \mathbb{P}_x-a.s.\,\forall x \in \overline{D}, $$ where $B_t$ is the usual standard $n$-dimensional Brownian motion.
We now that the generator for $X_t$ is the usual Laplace operator $\Delta$ with domain consisting the functions $f \in C^2(D)$ and $ \nabla f(x)\cdot n(x) = 0$ on $\partial D$.
However, by the Skorohod representation of $X_t$, for any $C^2$ function $f$, we may use It$\hat{o}$'s formula to get $$ f(X_t) = f(X_0) + \int_0^t\nabla f(X_s)\cdot dB_t + \int_0^t \nabla f(X_s) \cdot n(X_s) dL_s +\frac{1}{2} \int_0^t\Delta f(X_s)ds,\quad \mathbb{P}_x-a.s\, \forall x \in \overline{D}. $$ Here $\nabla f$ might not be zero on $\partial D$. Now we fix such $f$, the question is, is there a non-trivial measure $\nu$ on $D$ so that $\mathbb{E}_\nu[f(X_t)]$ is constant in time? If there is, how to find it?
My thought: it seems like the only thing we can do here is to solve for $\nu$ in the following problem: $$ \lim_{t\downarrow 0}\frac{1}{t}\int_D P_tf(x) - f(x) \nu(dx) = 0, $$ for which $P_t$ is the semi-group of $X_t$. Since $f$ might not be in the domain of the generator, and $\frac{d}{dt}\mathbb{E}_x[\int_0^t \nabla f(X_s)\cdot n(X_s)dL_s]$ at $t = 0$ is not defined pointwise for $x$ if $f$ itself is not Neumann. But it does not seem like such $\nu$ solves the problem.
Thanks in advance.