  We report on the study of the phase dynamics of high critical temperature
superconductor Josephson junctions. We realized YBa$_2$Cu$_3$O$_{7-x}$ (YBCO)
grain boundary (GB) biepitaxial junctions in the submicron scale, using low
loss substrates, and analyzed their dissipation by comparing the transport
measurements with Monte Carlo simulations. The behavior of the junctions can be
fitted using a model based on two quality factors, which results in a frequency
dependent damping. Moreover, our devices can be designed to have Josephson
energy of the order of the Coulomb energy. In this unusual energy range, phase
delocalization strongly influences the device's dynamics, promoting the
transition to a quantum phase diffusion regime. We study the signatures of such
a transition by combining the outcomes of Monte Carlo simulations with the
analysis of the device's parameters, the critical current and the temperature
behavior of the low voltage resistance $R_0$.
