The electrical conduction mechanism of the thin (Ba,Sr)$TiO_3$ film capacitor has been investigated at low field region (<200kV/cm). To reveal that mechanism, current-time measurement was performed. With that method, the two different currents - dielectric relaxation current and leakage current -are separated. For the purpose of setting up the conduction model, we varied the measurement temperature, the strength of electric field, and the polarity of electric field. From this result, we observed that the relaxation current has the same dependency of the Hopping process, leakage current, and of the Poole-Frenkel process. Hence, these two different currents seem to come from the conduction -"trap". In general, post-annealing process has an effect of removing the trap of devices. To observe the origin of the trap, we employed the Rapid Thermal Annealing (RTA), and Oxygen Plasma annealing process. At oxygen atmosphere (@1 atm ,600℃), the RTA process reduced the level of the two types of current, however at 20 mtorr vacuum, in the same temperature , RTA did not show the annealing effect. This data revealed the origin of the trap -oxygen vacancies in the thin BST film. To support this postulation, oxygen plasma annealing (@20mtorr,450℃) was carried out. The plasma annealing showed the similar effect with the RTA of oxygen atmosphere (@1 atm ,600℃). The aforesaid results suggest that electrical conduction mechanism of the thin BST film capacitor at low field region is originated from the trap of oxygen vacancies.