Irreducibility and invariant measure of diffusions

Question

I am reading up the textbook Stochastic Analysis and Diffusion Processes written by Gopinath Kallianpur and P. Sundar, and am coming up with some questions. Here is the context.

We consider the solution of the $d$-dimensional SDE $$dX_t=b(X_t)dt+dW_t,$$ where $b$ is a smooth vector field on $\mathbb{R}^d$ such that the solution does not explode. And here are my questions:

1. In Remark 11.3.1 there is a statement saying that if $\mu$ is an invariant measure of the process $X$, then $\mu$ should have a density.
2. Let $p_t(x,dy)$ be the transition kernel of the process $X$. And assume that the process has an absolutely continuous invariant measure. In the proof the author asserts that the kernel has a density $p_t(x,y)$ and we even have $p_t(x,y)>0$ for almost every $x,y \in \mathbb{R}^d$.

I am not able to justify these two statements. Could anyone help?