The battery will discharge as needed to supply the current demand of the load impedance that you are using. The trick in the joule thief circuit is that it uses a technique that allows a battery to discharge to a much lower state than usual. While a 1.5 volt battery would usually be considered unusable at 1 volt the joule thief circuit continues to use the battery down to a level of perhaps 0.3 volts. Thus the name implies that you are able to steal a bit more energy from the battery.
This question is really above my technical ability to accurately answer. I have a feeling that it would have to be drastically changed and redesigned to operate at that frequency. The question would have to include what your application is and what are the other parameters of what you are trying to accomplish. The joule thief as designed seems to be a power circuit to use the energy in a battery to drive a small load like an LED. None of these parameters would require a 1 gHz oscillator. Keep in mind also that most modern high energy density batteries do not tolerate a very deep discharge and would be damaged if left on an joule thief circuit to the final fail voltage.
This question is really above my technical ability to accurately answer. I have a feeling that it would have to be drastically changed and redesigned to operate at that frequency. The question would have to include what your application is and what are the other parameters of what you are trying to accomplish. The joule thief as designed seems to be a power circuit to use the energy in a battery to drive a small load like an LED. None of these parameters would require a 1 gHz oscillator. Keep in mind also that most modern high energy density batteries do not tolerate a very deep discharge and would be damaged if left on an joule thief circuit to the final fail voltage.
