  We compute explicitly, and without any extra regularity assumptions, the
large time limit of the fibrewise heat operator for Bismut-Lott type
superconnections in the L^2-setting. This is motivated by index theory on
certain non-compact spaces (families of manifolds with cocompact group action)
where the convergence of the heat operator at large time implies refined
L^2-index formulas. As applications, we prove a local L^2-index theorem for
families of signature operators and an L^2-Bismut-Lott theorem, expressing the
Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes.
With slightly stronger regularity we obtain the respective refined versions: we
construct L^2-eta forms and L^2-torsion forms as transgression forms.
