Exsistence and uniqueness of stationary density for Markov Chain

Question

Suppose we're given a function $f:\mathbb{R}^2\to\mathbb{R}$. We define a Markov Chain $(X_n)$ by \begin{align} X_0&\sim f_X, \\ X_n&=f(X_{n-1},Y_{n-1}), \end{align} where $(Y_n)$ is a sequence of iid random variables with some density $f_Y$ ($Y_n$ and $X_0$ ar also independent). We're searching for a stationary density $f_X$. Are there any theorems which could give information about the existence or the uniqueness of such a density? While there are a lot of results concerning such a problem with discrete distribution of $X_n$ and $Y_n$, the situation when the distributions are continuous seems hard to find. Or maybe I'm missing something obvious?