I got confused with this Markov Chain problem:
suppose the kernel $Q$ is $Q_x=N(cx,1)$, $c$ is a fixed constant with $|c|<1$ and the stationary distribution is $\pi=N(0,\frac{1}{1-c^2})$.
I want to verify the stationary distribution, so I have $$\pi Q = \int_{x \in \mathbb{R} } Q_x \, d\pi(x) $$
The problem is, how to evaluate that integral with respect to the measure $\pi(x)$? In other words, how to figure out this integral?
Thank you.