These "look-alikes" have related but different meanings. To persecute is to harass, to torment, to treat badly, to bother, to worry, to oppress, to trouble: "Some teen-agers feel that their parents persecute them." "Businessmen sometimes think that government agencies are set up solely to persecute them." Prosecute, primarily a legal term, means "to seek, force, or obtain by a legal process," "to start or conduct legal proceedings against." It also means "to carry forward some action already begun." "If you commit theft and are caught, you will be prosecuted." "Now that war has begun, the government will prosecute it vigorously."
Dictionary of problem words and expressions. Harry Shaw. 1975.