let $X_t$ be an Ito diffusion with drift $\mu(x)$ and noise $\sigma(x)$ such that there exists a stationary distribution $\rho_0(x)$ for the process.
suppose that I want to calculate $$\mathbb{E}\left[\int_0^h f(X_t)df(X_t)\right]$$ where here the expectation is taken both with respect to realizations of the process and with respect to the initial condition. suppose $f$ is $C^{\infty}(\mathbb{R})$.
1- In the particular case when the initial condition is itself sampled from $\rho_0$, is there a straightforward solution to the above?
2- Is it possible to rewrite this expectation in terms of the transition probability $p_t(y|x)$ that solves the forward Kolmogorov equation?
I apologize if this is obvious, I am a beginner in stochastic calculus.