No, your Wife cannot just kick you out of the marital residence, especially not if your name is on the title or the mortgage. Even if it's not, it is still the marital residence and you have a right to be there. And no, there is no "abandonment" in Florida, per se. However, even if you leave you will still need to assist with the bills and expenses for the children. As for whether you should stay, I would need more information in order to properly answer that question. Generally speaking, if you want to stay in the house and have the house awarded to you in the event of a subsequent divorce, you should probably stay there. However, there may be multiple reasons why it is more advisable for you to leave.
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Please note that THESE COMMENTS ARE NOT INTENDED AS LEGAL ADVICE and are for informational purposes only. This response is not intended to create any attorney-client relationship and is only based on the limited facts given. The response might change should additional facts be learned and should not be relied on as legal advice. It is recommended that you consult with an attorney who can properly assess the situation, as well as all pertinent facts, prior to taking any action based on the foregoing statements.
Unless your wife has a court order that gives her exclusive use and possession of the home she cannot legally keep you from the house. It is no different than if the two of you had a fight and plan to reconcile. Unless there is an injunction in place or some other order saying that she has exclusive use and possession you are not prohibited from that place. I do think you need legal advice on this issue, as this communication is not entirely sufficient.