In a similar vein to my previous question , can a mark from a Fighter, Battlemind, etc. be maintained when the target succeeds on a Stealth check vs. a Defender's passive Perception and becomes Hidden with respect to that Defender?
Stealth isn't a defined condition in 4e. Using the stealth action, on a success, makes the character Invisible and Hidden. For details on the difference, see this question.
These conditions inherently have no special effect on Marking. If you were marked when you become invisible or hidden or both, you are still marked and it ends however it normally would.
In the case of a Fighter who needs to attack to maintain his mark, Invisibility wouldn't do much because he can still attack the target (at a -5 penalty, but a Fighter doesn't need to hit). Hidden, on the other hand, means the fighter does not know where the target is, so attacking to maintain the mark is much more difficult.
The Rules Compendium (p. 221) gives some suggestions for what to do when you cannot see a creature and want to attack it. One of those is to pick a square you think the creature is in, and target your attack there. If you guess correctly the mark could be maintained.
Marks aren't maintained and only end in the following circumstances:
Absent the above situations the marked condition last for as long as the power or effect says that it does.
Making a stealth check and going hidden will not prematurely end a mark. If a mark is scheduled to end and the creator of the mark would like to mark the character again then it must fulfill the conditions of whichever power it wishes to use to mark the target.
This just drops down to the basic question of "Can I target a hidden creature?" and the answer (in general) is yes. From the Rules Compendium, pg 105:
If you don't have line of effect, it doesn't matter if the target is hidden or not, the target is immune to nearly all powers. If you can't see the target, then some powers that mark cannot be used. Those are pretty rare. The only at-will powers that can be foiled by invisibility (and hence hidden) are augmented Battlemind powers.
Remember that, unless the hidden creature does a double move (one move to become hidden and another move while hidden), everyone does indeed know where the hidden character is. A DM could consider the Intelligence of a monster but everyone else will know that the character is located exactly where they were last seen. Check the Rules of Hidden Club.