Understanding Convergence of Transitional Probabilities via Krylov Bogolivob Theorem

Question

Could anyone break and write a few more lines for me of the proof of the Theorem 1.10 here, by Prof. Martin Hairer, on page 7 in particular. I am not understanding what he means by (1) the continuity of $\mathcal{P}\psi$, (2) why the same lines are repeated twice, (3) how $\mathcal{P}^N$ came out, (4) how a factor $2$ comes at the end line, $(5)$ What is the action of $\mathcal P$ on $\psi$ exactly, $(6)$ What is action of $\mathcal P$ on $\mu^*$ and then on $\psi$? It would be really great if someone helps me to understand these points.

Answer 1

1. By the Feller property, the function $\mathcal P\varphi$ is continuous. Since $\varphi$ is assumed to be bounded, then $\mathcal P\varphi$ is also bounded. Therefore, by the weak convergence of $(\mu_{N_k})_k$ to $\mu_*$, $\mu_{N_k}(\mathcal P\varphi)\to \mu_*(\mathcal P\varphi)$.
2. Probably a typo.
3. It should actually be $\mathcal P^{N_k}$. This follows from the definition of $\mu_N$; by rewriting this with $N=N_k$, we get a telescopic sum.
4. We get $$\lVert \mathcal P^{N_k}\varphi-\varphi\rVert\leqslant \lVert \mathcal P^{N_k}\varphi\rVert +\lVert\varphi\rVert$$ and from the fact that for all continuous bounded function $f$, $\lVert \mathcal Pf\rVert\leqslant \lVert f\rVert$, it follows that $\lVert \mathcal P^{N_k}\varphi\rVert\leqslant \lVert \varphi\rVert$.