Me and my friend were running late for a movie, and were running through the mall to the theatre. When we ran past two mall security 'officers', they told us to stop running. Being foolish; we started running again, and the security people just kept walking. When we went up the escalator to the theatre, two security people were behind us, and we were forced to go back down the escalator toward the exit. There was an additional mall cop by the doors, and we said we wanted to leave. He let us go, but said that 'running in the mall was not recommended'. We thought this was the end of it, but all 3 mall cops ended up following us after we left. We started walking home; along the path beside the mall (still their property). As we passed additional various mall exits, some mall cops with bikes cut us off. We told them we were just leaving and they said 'thats bull'. They continued to be very rude and used vulgar language. We were questioning why they were mad at us still; and while we didn't do this in the upmost polite fashion, we still were fairly polite. (At this point we were just goofing around a bit.) They told us we were 'banned' and that we had to leave the property, or else we would be subject to 'arrest for trespassing'.

What can I do? I like this mall and want to go back... They didn't specify how long we were 'banned' for, and they (in my opinion) were fairly rude and powertrippy...
All we did is run...

It is private property and they have the right to ban you from it. If they didn't take your name or photos or whatever, they probably won't remember who you are if you go back.

I'd just give it some time - I'd wait several weeks. When you go back, go as a customer when you actually plan to eat there or buy something in a store - not just to hang out and loiter with your friends. Better yet, go with a parent. In the future, don't act up or they will probably arrest you. Being a mall cop has got to be a pretty boring job - don't give them the chance to have fun at your expense.

So, essentially, you want to know if they can prohibit you from returning to the mall, ever? Assuming the speak for the management of the mall, in all likelihood, yes.

How old are you?

You were running through the mall, were told to stop running, continued running and think mall security was rude?

And then you left by bicycle? How old are you?

You are banned until you are told you are not banned.

In the US anyway, stores and malls and restaurants can throw out and 'ban' anyone they want for any reason that doesn't fall under Civil Rights discrimination (race, creed, origin, gender, etc). Sometimes of course the lines are blurred, and those cases end up in court. But I don't see that you have a case.
Running not only is a risk to others walking, or you injuring yourselves and suing, but also strongly suggests that you are running FROM something. They don't have to prove you stole or did any other criminal act.