Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence:

$ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $

How can we find explicitly the stationary distribution of $X_n$? Or at least some moments of it?

**Edited**: In my case $ F = X_n - \Phi(\alpha X_n) $ (where $\alpha$ is a constant and $\Phi$ is CDF of the standard normal distribution) and $W_n$ has Bernoulli distribution with success probability $p$.