Uma Thurman and Arpad Busson Face Off over Custody of Their Toddler

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Uma Thurman's former fiancé, French financier Arpad Busson, has filed suit in Manhattan over custody of their 2-year-old daughter, Luna.

The actress, who lives in New York City, began dating the London-based multimillionaire in 2007 after meeting at a dinner in Milan, two years after she and actor Ethan Hawke divorced. She has two children with Hawke – Maya, 16, and Levon, 12.

"It is unfortunate that this very private discussion regarding Mr. Busson's visitation rights and his participation in decision making has been made public," a spokesperson for Thurman tells PEOPLE in a statement. "However, we are optimistic that a fair agreement for both sides will be reached out of court."

Busson has two sons – Flynn, 16, and Cy, 11 – with supermodel Elle Macpherson.

The on-again, off-again couple were engaged in 2008, and split briefly in 2009 before welcoming their daughter in July 2012 – but broke up for good earlier this year.

Court papers show the custody case was assigned to matrimonial court Tuesday and records will be sealed.

Busson's lawyer had no comment. Thurman's attorney did not immediately return calls for comment.

Busson, meanwhile, has been linked recently to The English Patient actress Kristin Scott Thomas. The pair was first spotted in London in May and then holding hands in Paris in July.

* With reporting by JULIE JORDAN

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