It’s quite common for the house that you live in to be in only one person’s name – and sometimes that person is your ex-spouse. The two of you may have done that for asset protection purposes, or other reasons.
Just because the house is in your ex-spouse’s name, that doesn’t allow your ex-spouse to just kick you out of the house, or force you out of the house. You are entitled to live there, just as your ex-spouse is.
In some situations, especially where there has been domestic violence or threats against you or your child, you may apply to the Family Court for you to have sole occupation of the house. This is usually only made in exceptional circumstances. The court must consider things like the needs of the parties and whether such an order is in the best interests of the child, and balance this with the rights of your ex-spouse.
If you do however fear violence, you should immediately seek legal advice or talk to the police. The Family Court can sometimes be slow when processing an application. At any rate, even if you do decide to leave, this will not affect your right to a share of the property.