  We prove the continuity and strict positivity of the multi-layer extension to
the stochastic heat equation introduced in [OW11] which form a hierarchy of
partition functions for the continuum directed random polymer. This shows that
the corresponding free energy (logarithm of the partition function) is well
defined. This is also a step towards proving the conjecture stated at the end
of the above paper that an array of such partition functions has the Markov
property.
