As I am licensed in Florida and Vermont, I cannot offer you specific legal advice as to New York State law. However, generally, the situation you describe appears to be a hit and run situation. You may have coverage under your Uninsured Motorist coverage on your own vehicle. Also, as you probably know, there would be coverage under any collision coverage which you purchased for your vehicle. You should advise your own carrier of this incident if you've not done so already.
If you meant to ask "Is their insurance liable", you would need to prove that either the insured or someone given permission to drive the insured vehicle caused the damage.
The answer would depend on your coverages, whether the car was really stolen, and whether NY law requires insurance companies to pay collision claims as the first party insurance carrier (yours) or as the 3rd party carrier (the car the hit you). Search this site for a NY lawyer or pull out your yellow pages and call a personal injury lawyer in your area for the answer. In my state, Michigan, your own carrier would pay collision coverage, subject to your deductible and you may be able to get the deductible waived or repaid. I do not know how the NY system works, but the local lawyer will. Call him/her, right away.
Your insurance may cover the loss and, given the stolen car issue, may be the fastest way of repairing your car. The other driver's insurance company may cover the property damage if it is proven that the car was not really stolen. Many people report their vehicles stolen after they commit a hit and run car accident.