It's not just enough to know its yours and verify it by having the serial number

You have to prove you are the rightful owner of that serial number

What's stopping the other person from saying you sold him the bike last year for cash? No receipt. No paper trail.

How can you prove you ever owned that serial number in the first place?

See what I'm saying?

Without some proof of your loss the cops are going to side with the person who is in possession of the bike today.

Did you file a police report when it was stolen? That's the kind of thing that would give you credible documentation that the bike was stolen and not sold. Even then the current owner can claim you sold it then filed a police report. Though at that point there is a burden of proof that starts to spread to the current owner.

It's just a tough thing to prove.