Do your pets' health go back up on their own?

Say, you lose health during a battle and don't have money to heal; does your pet's health go back up on its own if you wait long enough, like how its mana goes down?

@Edibbs, why yes, it does. o: But it recovers at a waaay slower rate than how mana depletes.
Like you may have to wait an hour to notice a full recovery. o:

@Flute

Thanks for responding! I was wondering because I was running out of money to give to the Herbalist. e.e