Legally you can't stop him, but you can tell him to put up or shut up. If it continues I would recommend you seeing an attorney to take over and inform him to put up or shut up. At some point it could get into abusive use of conflict, but not yet.
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NO. He has a right to file a petition -- you always have the option of ignoring him.
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Consult with an attorney and, depending on his basis for the claim, think of two options: (a) ignore him and have an attorney tell him you will not be responding to these emails anymore, or (b) file first and get the drop on him.
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Ignore him, but if it gets really bad consider talking to an attorney for filing an anti harassment order. There may be enough there to meet the elements of harassment . Besides that he has the right to file for modification, but that does not mean he will get it. If he does, you might want to hire an attorney.