We prove the continuity and strict positivity of the multi-layer extension to the stochastic heat equation introduced in  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.
"Continuity and strict positivity of the multi-layer extension of the stochastic heat equation." Electron. J. Probab. 25 1 - 41, 2020. https://doi.org/10.1214/20-EJP511