  We study mesoscopic fluctuations in the supercurrent of a Josephson junction
consisting of a topological insulator microbridge between two conventional
superconductors. In the model, we account for the strong proximity effect when
superconductors induce a gap in the spectrum of surface states as well as a
magnetic field piercing the junction area that causes depairing and gap
filling. The overall magnitude and functional form of the Josephson current
fluctuations are determined analytically, and found to sensitively depend on
the coupling strength to surface states, Thouless energy, and pair-breaking
energy scales in the problem. We also study the density of states that can be
measured by scanning probes. Technically, mesoscopic fluctuations on top of the
mean field description of the proximity effect in the topological region are
described by a field theory approach, the replica nonlinear $\sigma$-model in
the class-D of the extended symmetry classification.
