ergodic theorem for expectation of positive recurrent diffusion

Question

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{t\to\infty}\frac{1}{t}\int_0^t f(X_s)ds=\int_{-\infty}^\infty f(x)\mu(dx)$$ almost surely whenever $f\in L^1(\mu)$. I'm looking for sufficient conditions on $f$ such that a similar limit theorem holds whenever $X_0$ is deterministic: $$\lim_{t\to\infty}E\left[f(X_t)\right]=\int_{-\infty}^\infty f(x)\mu(dx).$$ Since the distribution of $X_t$ converges weakly to $\mu$, it is enough to have $f$ bounded and continuous. However, I'm interested in cases where $f$ is allowed to be unbounded and have full support. Would $f$ being in $L^2(\mu)$ suffice?