  We develop a theory for energy and spatially resolved tunneling spectroscopy
of topological quantum spin Hall helical states driven out of equilibrium. When
a helical liquid is constrained between two superconducting reservoirs
transport at the edge is governed by multiple Andreev reflections. The
resulting quasiparticle distribution functions of the edge channels exhibit
multiple discontinuities at subgap energies with the periodicity of an applied
voltage. The combined effect of interactions, disorder, and normal scattering
off the superconducting interface leads to the inelastic processes mixing
different helicity modes, thus causing smearing of these singularities. If
equilibration is strong, then the distribution functions of the edge channels
tend to collapse into a Fermi-like function with an effective temperature
determined by the superconducting gap, applied voltage, and intraedge
interaction parameter. We conclude that mapping out nonequilibrium distribution
functions may help to quantify the relative importance of various relevant
perturbations that spoil ideally ballistic edge transport.
