  We investigate classical thermal activation (TA) and macroscopic quantum
tunneling (MQT) for a YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) Josephson junction
coupled to an LC circuit theoretically. Due to the coupling between the
junction and the LC circuit, the macroscopic phase dynamics can be described as
the escape process of a fictitious particle with an anisotropic mass moving in
a two-dimensional potential. We analytically calculate the escape rate
including both the TA and MQT regime by taking into account the peculiar
dynamical nature of the system. In addtion to large suppression of the MQT rate
at zero temperature, we study details of the temperature dependece of the
escape rate across a crossover region. These results are in an excellent
agreement with recent experimental data for the MQT and TA rate in a YBCO
biepitaxial Josephson junction. Therefore the coupling to the LC circuit is
essential in understanding the macroscopic quantum dynamics and the qubit
operation based on the YBCO biepitaxial Josephson junctions.
