In chap 12 of Stoock and Varadhan Multidimensional diffusion processes in section 12.2 markov selections page 290 one reads
I couldn't find an example that fit the conditions (a)-(d).
One would guess that when a there is a unique strong solution then conditions (a) - (d)
I can't see why condition (b) and (c) hold in the above case.
It is indeed, (c) is the Markov property, You can find a good explanation at chap 6 (for the not so familiar product $\delta_{(\tau(\omega),x(\tau(\omega),\omega)) } \otimes_{\tau(\omega)} Q_\omega$ - see pg 139) and (b) is just the translation of the trajectory that would begin at zero to a trajectory that is constant up to $s$ and then begins to move. Note that time independence is being used here (see lemma 6.5.1).
See chap 1 for regular conditional probability distributions, page 34.
