Let $E$ be a LCH second countable topological space and let $\mathcal{E}$ be its Borel $\sigma$-algebra.

Let $(P_t)_{t \geq 0}$ be a conservative transition function on $(E, \mathcal{E})$.

This means by definition that, $\forall t \geq 0$, $P_t : E \times \mathcal{E} \mapsto [0,1]$ such that:

$\forall t \in [0,\infty)$, $P_t(x,\cdot)$ is a probability measure on $E$ $\forall x \in E$

$\forall t \in [0,\infty)$, $P_t(\cdot, A)$ is $\mathcal{E}$-measurable $\forall A \in \mathcal{E}$

$\forall t,s \in [0,\infty)$, $P_{t+s}(x,A) = \int_E P_t(x,dy) P_s(y,A)$, $\forall (x,A) \in E \times \mathcal{E}$

Assume further that $P$ is normal, i.e. $P_0(x,\cdot) = \delta_x \quad \forall x \in E$.

The so-called Feller property (as in Rogers and Williams and Revuz and Yor, some authors replace $C_0$ with $C_b$, I will probably ask on this point in another post) now reads as follows:

1 - for all $f \in \mathcal{C}_0(E)$, $P_t f \rightarrow f$ uniformly as $t \rightarrow 0$.

2 - for all $f \in \mathcal{C}_0(E)$, $P_t f \in \mathcal{C}_0(E)$.

I would like to have (many) examples of conservative transition functions that satisfy 1 but not 2. I would like to have both trivial (if possible) and non-trivial examples. If possible some references to existing literature are welcome, but I noted that classical books on Markov processes do not exhibit examples of processes lacking the Feller property.

Point $2$ can fail to hold in many ways: the most brutal is that $P_t f$ is not even continuous for some continuous $f$ vanishing at infinity. But we can also have that $P_t f$ is continuous (or even continuous and bounded) for each $f \in \mathcal{C}_0(E)$ but fail to vanish at infinity for some $f$. If possible, I would like to have all sort of examples.

Thanks a lot in advance for your answers.