Markov processes lacking the Feller property

Question

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.

Answer 1

Here is a not so trivial example: take for $P_t$ the transition semigroup for the SDE $$ dx = -x^3\,dt + dW\;, $$ on $\mathbf{R}$. This is not Feller in your sense because solutions come in from infinity in finite time, so $P_t f$ fails to be in $C_0$ even if $f$ is. It is however Feller in the other definition you've seen in the literature, which causes no end of confusion...

Answer 2

A trivial example. Let $E = [0,\infty)$. Consider the translation semigroup for $c >0$: \begin{align} P_t f(x) &= f(x+ct),\, x >0, \\ P_t f(0) &= f(0). \end{align} So the particle moves to the right with speed $c$ if the starting point is not zero, otherwise it stays in zero.

Since $f \in C_0$: for any $ \varepsilon >0$ find $\delta(\varepsilon) >0$ s.t. $|f(x) -f(y) | \leq \varepsilon$ for $|x-y| \leq \delta$. Then $|P_tf(x) -f(x)| = |f(x+ct)-f(x)| \leq \varepsilon$ as long as $ct < \delta(\varepsilon)$. However, $$ \lim_{x\to 0} P_tf (x) = \lim_{x\to 0} f(x+ct) = f(ct) $$ which is not always equal to $P_tf(0)=f(0)$. So this is an example, where (1) holds, but (2) does not. It can be modified to diffusions (to get a non-deterministic example) which do not hit the boundary point $0$.