Looking for an example of a process that holds the Markov property but doesn't hold the strong Markov property

Question

I am desperately looking for a Markov process which does only hold the Markov property but doesn't hold the strong Markov property. All examples I can think of hold the Markov property, as well as the strong Markov property. Could some one maybe give me an example of a process that does hold the Markov property but doesn't hold the strong Markov property?

Answer 1

There are three voted-up examples at Mathoverflow. The accepted example from George Lowther said

Consider the following continuous Markov process $X$, starting from position $x$

1. if $x = 0$ then $X_t=0$ for all times.
2. if $x \not = 0$ then $X$ is a standard Brownian motion starting from $x$.

This is not strong Markov (look at times at which it hits zero).