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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.

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