adverb Perhaps; indicates that the proposition may be true (is not certainly false) regardless of any facts or circumstances known to, stated by or implied by the speaker.
Possibly, they will make gains in the midterm elections.
adverb In the realm of possibility; indicates that the action may successfully be performed (is not impossible) regardless of any facts or circumstances known to, stated by or implied by the speaker that might limit the performance.
I'm much stronger than you, so you can't possibly win.