Persecute and prosecute are occasionally mixed up. Persecute means to torment or cause suffering, usually because of some perceived difference. School bullies persecute those whom they perceive to be weaker than they are. In the adult world, people may be persecuted because of their religious beliefs, political beliefs, ethnicity, sexual orientation or social grouping.
Prosecution, on the other hand, involves bringing legal charges against someone. In some US states jaywalkers are prosecuted. That means that they face court charges because they crossed the road at the wrong time. And these days, if you have been persecuted for your beliefs, you are often able to prosecute your persecutor.