To whom does the Tallit belong?

Last year, I was the Baal Koreh in my yeshiva, and I needed a tallit. I made an agreement with a third-year student who was a Kohen and had a tallit, that I would fold his tallit, and in return, I could use it on Mondays and Thursdays. Since then, I developed a sort of partial ownership of the tallit, and it turned out that I even used it more. At the end of the year, he collected phone numbers from several guys to contact them, including mine. I reminded him several times to take his tallit, but he didn't. Ultimately, the tallit has remained with me for 3 months now, and he never called to ask for it. He lives in another city, I have no idea where, and I tried to get his phone number but couldn't. Can the tallit now be officially considered mine, as if the owner has given up on it? He was a Breslov Hasid, and this has certain implications. Am I allowed to replace the tzitzit as if it were mine, or do I need to turn the world upside down to find him and not touch the threads of the tallit according to the law?