Before revision:  The noisy fluctuations arise from the synthesis reaction rate . We compute two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP), in order to examine the likelihood for transcriptions: The mean time scale for the system exits the low concentration state (the longer the exit time, the less likely for transcription) and the switch probability from low concentration states to high concentration states (corresponding to likelihood for transcription) .
After revision:  The noisy fluctuations arise from the synthesis reaction rate is under additive and multiplicative asymmetric stable L\'evy fluctuations .