Continuity and strict positivity of the multi-layer extension of the stochastic heat equation

TL;DR

This paper proves the continuity and strict positivity of the multi-layer extension of the stochastic heat equation, establishing the well-definedness of the free energy and advancing the understanding of its Markov properties.

Contribution

It demonstrates the continuity and positivity of the multi-layer stochastic heat equation extension, a key step in analyzing the associated continuum directed random polymer.

Findings

Proved continuity of the multi-layer stochastic heat equation extension.

Established strict positivity of the solution.

Supported the conjecture on the Markov property of partition function arrays.

Abstract

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.