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We characterize topological features of Josephson junctions formed by coupledmesoscopic chiral $p$-wave superconducting islands. Through analytic andnumerical studies of the low-lying BdG (Bogoliubov-de Gennes) spectrum, weidentify localized MBS (Majorana bound states) nucleated in Josephson vorticesby the application of a perpendicular magnetic field. Additionally, wedemonstrate the existence of an extended MBS that is delocalized around theouter perimeter of the coupled islands, which has measurable consequences onthe Josephson supercurrent and phase dynamics of the junction. In particular,we predict a change in the critical current diffraction pattern in which theodd integer-flux nodes are lifted in a fermion parity-dependent fashion. Wemodel the competing stochastic effects of thermal noise and macroscopic quantumtunneling within the RCSJ framework and show the emergence of a bimodalcritical current distribution. We demonstrate that increasing the paritytransition rate suppresses the bimodal nature of the distribution, thusstrongly emphasizing the non-trivial parity dependent nature of the many-bodyground state. Finally, we consider a trijunction geometry with three islandsand discuss possible schemes to braid Majorana bound states by moving theJosephson vortices to which they are bound.

Signatures of Majorana bound states and parity effects in two-dimensional chiral p-wave Josephson junctions