# Can a continuous time Markov processes have transition densities only at certain times?

Suppose that $$X=(X_t)_{t\ge 0}$$ is a Markov process on some state space $$E\subset\Bbb R^n$$ with transition semi-group $$(P_t)_{t\ge 0}$$. It is often nice to know, whether the transition measures have Lebesgue-densities. Usually when I see such conditions, they are assumed to hold for all $$t\ge0$$.

QUESTION: Is it possible that for some $$x\in E$$ and $$s,t\ge0$$ the measure $$P_t(x,\,\cdot\,)$$ is absolutely continuous with respect to Lebesgue's measure while $$P_s(x,\,\cdot\,)$$ is not? If so, what are interesting examples? Are there restrictions for the set of $$t\in(0,\infty)$$ for which $$P_t(x,\,\cdot\,)$$ is a.c.?

If $$X$$ is not homogeneous with respect to time, the corresponding phenomenon can of course occur. However, I am struggling to get an intuition for the homogeneous case, in particular when $$X$$ is the solution to an SDE.

Any idea or reference is much appreciated.