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We report the detailed study on the low temperature dielectric dynamics of the epitaxial BiFeO3 thin films grown on Nb-doped SrTiO3 substrate. The results indicate that the contributions from the thin film dominate the dielectric response, although it comes from both the thin film and the electrode interface. Furthermore, the origins of the low temperature dielectric anomalies are investigated with electric circuit fittings. A possible phase transition at 210 K is revealed from analysis with dielectric loss tangent. The dielectric constants obtained from the constant phase elements (CPEs) are more than 400 even at low temperatures. Finally, the physical significances of the CPE model are discussed.