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We investigate classical thermal activation (TA) and macroscopic quantumtunneling (MQT) for a Josephson junction coupled with an LC circuittheoretically. The TA and MQT escape rate are calculated analytically by takinginto account the two-dimensional nature of the classical and quantum phasedynamics. We find that the MQT escape rate is largely suppressed by thecoupling to the LC circuit. On the other hand, this coupling gives rise toslight reduction of the TA escape rate. These results are relevant for theinterpretation of a recent experiment on the MQT and TA phenomena in grainboundary YBCO Josephson junctions.

Theory of two-dimensional macroscopic quantum tunneling in a Josephson junction coupled with an LC circuit