Persecute has the following meanings:-
1.To constantly annoy, to harass, to afflict, or to bother someone: “Ever since the news of their break-up appeared in a newspaper, the reporters of other newspapers were after them to persecute them with odd questions.”
2. To treat someone cruelly or unfairly; especially, because of race or religious or political beliefs: “There is hardly any religious group that did not persecute the followers of other beliefs.”
Prosecute , on the other hand, has the following meanings:-
1. To initiate legal action as the result of a crime: “The police decided not to prosecute the protestors who raised slogans against the Chief Minister at the public rally.”
2. To follow through with something until it is finished: “The owner has the right to prosecute the trespassers.”
