Hi mpume,
What a great question!
The Mark reference you gave about when Jesus confronts the disciples about their argument took place in Capernaum, but there is also a verse in Matthew (4:12) that states when Jesus first started to his stay in Capernaum it fufilled a prophecy (Isaiah 8:23-9:1) (and kicked off his public ministry).
The best version I have found for this kind of question - the New Oxford Annotated Version- says that he "made his home" there and my other one says he "settled there." This wouldn't imply he necessarily owned the building.
From what I have read about the archives of the area, there is a house famous for being Peter's house, (but of course Matthew, Andrew, James, and John were all from there too.)
So yeah, I'll chime in with Chad and say He owns it all anyhow! And even though I can't answer your question for sure, because I don't know, a verse (also in Matthew - 8:20) did come to mind as soon as I read you post, "'Foxes have holes and the birds of the air have their nests, but the Son of man has nowhere to lay his head.'"
I'm curious to read if anyone knows anything more specific on this. It's definately going on my list of questions when I meet Him, like what He wrote in the sand.
fww