Did you just connect all the wires from the HID kit to the factory headlight connectors? If so the problem may be the OEM ground switching system, which has marginal current-carrying capacity and will have problems with the high startup current draw of the HIDs. Once they start and settle to their normal operating current they actually draw less than the stock bulbs, but proper connections are still important to keep them from going out.
I'd suggest that you wire the negative side of the ballasts directly to the battery negative terminal and completely bypass the factory ground wiring, this will make your lights much more reliable. If you still have problems after making this change, check all the wiring and connections to the HIDs (you should do this in any case) and make sure everything is secure.
If you do relocate the ground connections it will be absolutely necessary to disconnect the main DRL module by the glovebox, otherwise the DRL function will return.