TITLE: Markov chain with uncountable state space
QUESTION [7 upvotes]: I'm self-studying probability theory and struggling with understanding Markov chains on uncountable state spaces, notably I would like to solve the following exercise from this book.

$\textbf{15.2.3.}$ Consider the Markov chain on the real line $\mathbb{R}$, where $P(x, \cdot) = N(\frac{x}{2}, \frac{3}{4})$ for each $x \in \mathbb{R}$. Show that $\pi(\cdot) = N(0,1)$ is a stationary distribution for this chain. And how does one proceed in showing the chain is irreducible with respect to Lebesgue measure?
$P(x,A)$ is the probability that the chain will jump to a subset $A$ being at a point $x$.

In case of finite state space, I seem to have understood it as it boils down to solving a system of equalities. But I don't really see how it works when integrals are involved.
Could anyone offer some illuminating hints? I'll try to work out the rest myself.
Thanks in advance.

REPLY [4 votes]: Just for fun, let's prove the uniqueness of $\pi$ using 
characteristic functions. For the given transition kernel, 
 the characteristic function 
$\hat\pi$ of any invariant probability measure $\pi$ 
satisfies $$\hat\pi(t)=\hat\pi(t/2) \exp\left(-{t^2\over 2} [3/4]\right).$$
Substituting $t/2$ for $t$ in this equation, and repeating
we discover that, for every $n\geq 0$,  $$\hat\pi(t)=\hat\pi(t/2^n) \exp\left(-{t^2\over2} [1-1/4^n]\right).$$
Now let $n\to\infty$ and conclude that $\hat\pi(t)=\exp(-t^2/2)$, 
that is, $\pi$ must be a standard normal distribution.